University  of  California  •  Berkeley 


THE  THEODORE  P.  HILL  COLLECTION 

of 
EARLY  AMERICAN  MATHEMATICS  BOOKS 


,    ^ 


. 


3, 


Oat}    anb    ftljomson's    Series. 

A 

TREATISE 


OP 


PLANE  TRIGONOMETRY, 

AND  THE 

MENSURATION  OP  HEIGHTS  AND  DISTANCES. 

TO   WHICH   13   PREFIXED 

A  SUMMARY  VIEW  OP  THE  NATURE  AND  USB 
OP 

LOGARITHMS, 

ADAPTED  TO 

FHB  METHOD  OF  INSTRUCTION  IN  SCHOOLS  AND  ACADEMIES 


BY  JEREMIAH  DAY,  D.D.  LL.D. 

LATX  PRESIDENT  OF  TALB  COLLEGE. 


NEW  YORK: 
IVISON    &    PHINNEY,   178    FULTON  STREET; 

(SUCCESSORS  or  NEWMAN  fc  IVISON,  AND  MARK  H.  NEWMAN  *  co.) 
CHICAGO:  S.  C.  GRIGGS  A  CO.,  Ill   LAKE  ST. 

BUFFALO:    PHINNEY    t    CO.,    188    MAIN    STREET 

AUBURN:  J.  c.  IVISON  *  co.    DETROIT  :  A.  M'FARREN. 

CINCINNATI:  MOORE,  ANDERSON  *  co. 

1855. 


ENTERED,  according  to  Act  of  Congress,  in  the  year  1848,  by 
JEREMIAH    DAY, 

In  tla»  Clerk's  Office  of  the  District  Court  of  the  United  States  far  the 
Southern  District  of  New  York. 


THOMAS   B.    SMITH,    STERKOTYPER.  J.    D.   BEDFORD,   PRINTER, 

216  WILLIAM  STRKKT,  N.  Y-  138   FULTON    STREET. 


PLANE  TRIGONOMETRY. 


SCARCELY  any  department  of  Mathematics  is  more  impor- 
tant, or  more  extensive  in  its  applications,  than  Trigonometry.. 
By  it  the  mariner  traces  his  path  on  the  ocean ;  the  geogra- 
pher determines  the  latitude  and  longitude  of  places,  the  di- 
mensions and  positions  of  countries,  the  altitude  of  mountains, 
the  courses  of  rivers,  &c.,  and  the  astronomer  calculates  tha 
distances  and  magnitudes  of  the  heavenly  bodies,  predicts  the 
eclipses  of  the  sun  and  moon,  and  measures  the  progress  of 
light  from  the  stars. 

The  section  on  right  angled  triangles  in  this  treatise,  may 
perhaps  be  considered  as  needlessly  minute.  The  solutions 
might,  in  all  cases,  be  effected  by  the  theorems  which  are 
given  for  oblique  angled  triangles.  But  the  applications  of 
rectangular  trigonometry  are  so  numerous,  in  navigation,  sur- 
veying, astronomy,  &c.,  that  it  was  deemed  important,  to  ren- 
der familiar  the  various  methods  of  stating  the  relations  of  the 
sides  and  angles ;  and  especially  to  bring  distinctly  into  view 
the  principle  on  which  most  trigonometrical  calculations  are 
founded,  the  proportion  between  the  parts  of  the  given  tri- 
angle, and  a  similar  one  formed  from  the  sines,  tangents,  &c., 
in  the  tables. 

As  this  treatise  is  intended  to  form  a  part  of  Day  and 
Thomson's  Course  of  Mathematics  for  the  use  of  Schools  and 
Academies,  the  references  to  Algebra  are  made  to  Thomson's 
Abridgment ;  and  the  references  to  Geometry,  to  Thomson's 
Legendre,  as  well  as  to  Euclid's  Elements. 


CONTENTS. 


LOGARITHMS. 

Ftp 

SECTION    I.  Nature  of  Logarithms, 7 

II.  Directions  for  taking  Logarithms  and  their  Num- 
bers from  the  Tables, 14 

m.  Methods  of  calculating  by  Logarithms,  .... 

Multiplication,       22 

Division, 25 

Involution,        27 

Evolution, 29 

Proportion,       32 

Arithmetical  Complement, 33 

Compound  Proportion, 35 

Compound  Interest, 37 

Increase  of  Population,  ........  40 

Exponential  Equations, 45 

TRIGONOMETRY. 

SECTION  I.  Sines,  Tangents,  Secants,  &c., 47 

n.  Explanation  of  the  Trigonometrical  Tables,  .  .  59 
HI.  Solutions  of  Right  angled  Triangles,  ....  67 
IV.  Solutions  of  Oblique  angled  Triangles,  ...  85 
V.  Geometrical  Construction  of  Triangles,  ...  99 
VI.  Description  and  use  of  Gunter's  Scale,  ...  107 

VII.  Trigonometrical  Analysis,        116 

Application  of  Trigonometry  to  the  mensuration  of  heights  and 

distances, 130 

Notes, 148 


LOGARITHMS, 

SECTION  I. 

NATURE    OF    LOGARITHMS. 

ART.  1.  The  operations  of  Multiplication  and  Division, 
when  they  are  to  be  often  repeated,  become  so  laborious, 
that  it  is  an  object  of  importance  to  substitute,  in  their 
stead,  more  simple  methods  of  calculation,  such  as  Addition 
and  Subtraction.  If  these  can  be  made  to  perform,  in  an 
expeditious  manner,  the  office  of  multiplication  and  division, 
a  great  portion  of  the  time  and  labor  which  the  latter  pro- 
cesses require,  may  be  saved. 

Now  it  has  been  shown,  (Algebra,  189,  193,)  that  powers 
may  be  multiplied  by  adding  their  exponents,  and  divided, 
by  subtracting  their  exponents.  In  the  same  manner,  roots 
may  be  multiplied  and  divided,  by  adding  and  subtracting 
their  fractional  exponents.  (Alg.,  232,  239.)  When  these 
exponents  are  arranged  in  tables,  and  applied  to  the  general 
purposes  of  calculation,  they  are  called  Logarithms. 

2.  LOGARITHMS,  THEN,  ARE  THE  EXPONENTS  OF  A 

SERIES    OF    POWERS    AND    ROOTS. 

In  forming  a  system  of  logarithms,  some  particular  num- 
ber is  fixed  upon,  as  the  base,  radix,  or  first  power,  whose 
logarithm  is  always  1.  From  this  a  series  of  powers  is 
raised,  and  the  exponents  of  these  are  arranged  in  tables  for 
use.  To  explain  this,  let  the  number  which  is  chosen  for  the 


8  If  ATTTRE  OF   LOGARITHMS. 

first  power  be  represented  by  a.     Then  taking  a  series  of 
powers,  both  direct  and  reciprocal,  as  in  Alg.  163  ; 

a4,  a',  a9,  a1,  a°,  d~\  a"1,  a"9,  cT4,  &c. 
The  logarithm  of  a*  is  3,  and  the  logarithm  of  a~l  is — 1, 

of  a1  is  1,  of  cf  is— 2, 

of  a°  is  0,  of  of  is — 3,  <fec. 

Universally,  the  logarithm  of  a*  is  x. 

3.  In  the  system  of  logarithms   in  common  use,  called 
Briggs's  logarithms,  the  number  which  is  taken  for  the  radix 
or  base  is  10.     The  above  series,  then,  by  substituting  10 
for  a,  becomes 

104,    10s,    109,  101,  10°,  10~',  10-*,  10-*,  &c. 
Or  10000,  1000, 100, 10,  1,     -^,     rhr,  rsW,  &c. 

Whose  logarithms  are 
4,         3,         2,         1,         0,     —1,     —2,     —3,  <fec. 

4.  The  fractional  exponents  of  roots,  and  of  powers  of 
roots,  are  converted  into  decimals,  before  they  are  inserted 
in  the  logarithmic  tables.     See  Alg.  208. 

The  logarithm  of  a*,  or  a  °  • ' 3  8  8 ,  is  0.3333, 
of  eft,  or  a0-68'6,  is  0.6666, 
of  a*,  or  a0'4986,  is  0.4285, 
of  a^ora8-86'6,  is  3.6666,  <fec. 

These  decimals  are  carried  to  a  greater  or  less  number  of 
places,  according  to  the  degree  of  accuracy  required. 

5.  In  forming  a  system  of  logarithms,  it  is  necessary  to 
obtain  the  logarithm  of  each  of  the  numbers  in  the  natural 
series  1,  2,  3,  4,  5,  <fec. ;  so  that  the  logarithm  of  any  num- 
ber may  be  found  in  the  tables.     For  this  purpose,  the  radix 
of  the  system  must  first  be  determined  upon ;  and  then  every 
other  number  may  be  considered  as  some  power  or  root  of 


NATURE    07    LOGARITHMS.  ft 

this.  If  the  radix  is  10,  as  in  the  common  system,  every 
other  number  is  to  be  considered  as  some  power  of  10. 

If  the  exponent  is  a  fraction,  and  the  numerator  be  in- 
creased, the  power  will  be  increased  ;  but  if  the  denominator 
be  increased,  the  power  will  be  diminished. 

6.  To  obtain  then  the  logarithm  of  any  number,  according 
to  Briggs's  system,  we  have  to  find  a -power  or  root  of  10 
which  shall  be  equal  to  the  proposed  number.  The  exponent 
of  that  power  or  root  is  the  logarithm  required.  Thus 


0  .  84ft  1 


7=10 
20=10 
30=10 
400=109-6080 


1.8010 


.  4  T7  1 


of      7  is  0.8451 
therefore  the     I  of    20  is  1.3010 
logarithm        ]  of    30  is  1.4771 

.of  400  is  2.6020,  &c. 


7.  A  logarithm  generally  consists  of  two  parts,  an  integer 
and  a  decimal.  Thus  the  logarithm  2.60206,  or,  as  it  is 
sometimes  written,  2 +.60206,  consists  of  the  integer  2,  and 
the  decimal  .60206.  The  integral  part  is  called  the  charac- 
teristic or  index*  of  the  logarithm ;  and  is  frequently  omitted, 
in  the  common  tables,  because  it  can  be  easily  supplied, 
whenever  the  logarithm  is  to  be  used  in  calculation. 

By  art.  3d,  the  logarithms  of 

10000,  1000,  100,  10,  1,       .1,     .01,     .001,  &c. 
are        4,          3,         2,      1,    0,    -—1,    —2,      —3,  &c. 

As  the  logarithms  of  1  and  of  10  are  0  and  1,  it  is  evident, 
that,  if  any  given  number  be  between  1  and  10,  its  logarithm 
will  be  between  0  and  1,  that  is,  it  will  be  greater  than  0, 
but  less  than  1.  It  will  therefore  have  0  for  its  index,  with 
a  decimal  annexed. 

Thus,  the  logarithm  of  5  is  0.69897. 

*  The  term  index,  as  it  is  used  here,  may  possibly  lead  to  some  con- 
fusion in  the  mind  of  the  learner.  For  the  logarithm  itself  is  the  index 
or  exponent  of  a  power.  The  characteristic,  therefore,  is  the  index  of 
an  index. 


JO  NATURE    OF    LOGARITHMS* 

For  the  same  reason,  if  the  given  number  be  between 

10  and  100,       \  the  log.  (  1  and  2,  i.e.  1+the  dec.  part, 

100  and  1000,     >  will  be  )  2  and  3,        2+the  dec.  part. 

1000  and  10000,   )  between  (  3  and  4,        3-f-the  dec.  part. 

We  have,  therefore,  when  the  logarithm  of  an  integer  or 
mixed  number  is  to  be  found,  this  general  rule : 

8.  The  index  of  the  logarithm  is  always  one  less,  than  the 
number  of  integral  figures,  in  the  natural  number  whose  logo- 
rithm  is  sought :  or,  the  index  shows  how  far  the  first  figure 
of  the  natural  number  is  removed  from  the  place  of  units. 

Thus,  the  logarithm  of  37  is  1.56820. 

Here,  the  number  of  figures  being  two,  the  index  of  the 
logarithm  is  1. 

The  logarithm  of  253  is  2.40312. 

Here  the  proposed  number  253  consists  of  three  figures, 
the  first  of  which  is  in  the  second  place  from  the  unit  figure. 
The  index  of  the  logarithm  is  therefore  2. 

The  logarithm  of  62.8  is  1.79796. 

Here  it  is  evident  that  the  mixed  number  62.8  is  between 
10  and  100.  The  index  of  its  logarithm  must,  therefore, 
bel. 

9.  As  the  logarithm  of  1  is  0,  the  logarithm  of  a  number 
less  than  1,  that  is,  of  any  proper  frac tion,  must  be  negative. 

Thus,  by  art.  3d, 

The  logarithm  of  iV  or  .1  is  — 1, 
of  To7  or  .01  is  — 2, 
of  rsVo  or  .001  is  — 3,  &c. 

10.  If  the  proposed  number  is  between  T£T  and  io\0,  its 
logarithm  must  be  between  — 2  and  — 3.     To  obtain   the 
logarithm,  therefore,  we  must  either  subtract  a  certain  frac- 
tional part  from  — 2,  or  add  a  fractional  part  to  — 3  ;  that 


NATURE    OF    LOGARITHMS.  11 

is,  we  must  either  annex  a  negative  decimal  to  — 2,  or  a  pos- 
itive one  to  — 3. 

Thus,  the  logarithm 

of  .008  is   either — 2  — .09691,  or — 3  +  90309.* 

The  latter  is  generally  most  convenient  in  practice,  and  is 
more  commonly  written  3.90309.  The  line  over  the  index 
denotes,  that  that  is  negative,  while  the  decimal  part  of  th<* 
logarithm  is  positive, 

f  of  0.3,      is  T47712, 

The  logarithm    )  of  0.06,    is  J2777815, 

(  of  0.009,  is  3.95424, 

And  universally, 

11.  The  negative  index  of  a  logarithm  shows  how  far  the 
first  significant  figure  of  the  natural  number,  is  removed  from, 
the  place  of  units,  on  the  right ;  in  the  same  manner  as  a  pos- 
itive index  shows  how  far  the  first  figure  of  the  natural  num- 
ber is  removed  from  the  place  of  units  on  the  left.  (Art.  8.) 
Thus,  in  the  examples  in  the  last  article, 

The  decimal  3  is  in  the  first  place  from  that  of  units, 
6  is  in  the  second  place, 
9  is  in  the  third  place  ; 

And  the  indices  of  the  logarithms  are  1,  2,  and  3. 

12.  It  is  often  more  convenient,  however  to  make  the  in- 
dex of  the  logarithm  positive,  as  well  as  the  decimal  part. 
This  is  done  by  adding  10  to  the  index. 

Thus,  for  — 1,  9  is  written,       for  — 2,  8,  &c. 
Because  —1  +  10=9,  —2+10=8,  <fec. 

*  That  these  two  expressions  are  of  the  same  value  will  be  evident, 
if  we  subtract  the  same  quantity,  +.90309  from  each.  The  remainders 
will  be  equal,  and  therefore  the  quantities  from  which  the  subtraction  is 
made  must  be  equal. 


12  NATURE    OF   LOGARITHMS. 

With  this  alteration, 

i  T.90309  \  f  9.90309, 

The  logarithm  5  ¥.90309  >  becomes  j  8.90309, 

(  ¥.90309  )  (  7.90309,  &c. 

This  is  making  the  index  of  the  logarithm  10  too  great. 
But  with  proper  caution,  it  will  lead  to  no  error  in  practice. 

13.  The  sum  of  the  logarithms  of  two  numbers,  is  the 
logarithm  of  the  product  of  those  numbers  ;  and  the  differ- 
ence of  the  logarithms  of  two  numbers,  is  the  logarithm  of 
the  quotient  of  one  of  the  numbers  divided  by  the  other. 
(Art.  2.)   In  Briggs's  system,  the  logarithm  of  10  is  1.  (Art. 
3.)     If  therefore  any  number  be  multiplied  or  divided  by  10, 
its  logarithm  will  be  increased  or  diminished  by  1 :  and  as 
this  is  an  integer,  it  will  only  change  the  index  of  the  loga- 
rithm, without  affecting  the  decimal  part. 

Thus,  the  logarithm  of  4730  is  3.67486 
And  the  logarithm  of        10  is  1. 
The  logarithm  of  the  product       47300  is  4.67486 
And  the  logarithm  of  the  quotient     473  is  2.67486 

Here  the  index  only  is  altered,  while  the  decimal  part  re- 
mains the  same.     We  have  then  this  important  property, 

14.  The  DECIMAL  PART  of  the  logarithm  of  any  number  is 
the  same,  as  that  of  the  number  multiplied  or  divided  by  10, 
100,  1000,  &c. 

Thus  the  log.  of  45670,  is  4.65963, 

4567,  3.65963, 

456.7,  2.65963, 

45.67,  1.65963, 

4.567,  a65963, 

.4567,  T.65963,  or  9.65963, 

.04567,  ¥65963,       8.65963, 

.004567,  3765963,       7.65963 

property,  which  is  peculiar  to  Briggs's  system,  is  ot 


NATURE    OF   LOGARITHMS.  13 

great  use  in  abridging  the  logarithmic  tables.  For  when 
we  have  the  logarithm  of  any  number,  we  have  only  to 
change  the  index,  to  obtain  the  logarithm  of  every  other 
number,  whether  integral,  fractional,  or  mixed,  consisting  of 
the  same  significant  figures.  The  decimal  part  of  the  loga- 
rithm of  a  fraction  found  in  this  way,  is  always  positive.  For 
it  is  the  same  as  the  decimal  part  of  the  logarithm  of  a 
whole  number. 

IV.  If  a  series  of  numbers  be  in  GEOMETRICAL  progression, 
their  logarithms  will  be  in  ARITHMETICAL  progression.  For, 
in  a  geometrical  series  ascending,  the  quantities  increase  by 
a  common  multiplier  ;  (Alg.  359.)  That  is,  each  succeeding 
term  is  the  product  of  the  preceding  term  into  the  ratio. 
But  the  logarithm  of  this  product  is  t)  e  sum  of  the  logarithms 
of  the  preceding  term  and  the  ratio ;  that  is,  the  logarithms 
increase  by  a  common  addition,  and  are,  therefore,  in  arith- 
metical progression.  (Alg.  326.)  In  a  geometrical  progres- 
sion descending,  the  terms  decrease  by  a  common  divisor,  and 
their  logarithms,  by  a  common  difference* 

Thus,  the  numbers  1,  10,  100,  1000,  10000,  &c.,  are  in 
geometrical  progression. 

And  their  logarithms  0,  1,  2,  3,  4,  &c.,  are  in  arithmetical 
progression. 

•  See  Note  A. 
2 


14  THE   LOGARITHMIC    TABLES. 


SECTION   II. 

DIRECTIONS    FOR    TAKING    LOGARITHMS    AND    THEIR    NUM- 
BERS   FROM    THE    TABLES.* 

ART.  24.  The  purpose  which  logarithms  are  intended  to 
answer,  is  to  enable  us  to  perform  arithmetical  operations 
with  greater  expedition,  than  by  the  common  methods.  Be- 
fore any  one  can  avail  himself  of  this  advantage,  he  must 
become  so  familiar  with  the  tables,  that  he  can  readily  find 
the  logarithm  of  any  number ;  and,  on  the  other  hand,  the 
number  to  which  any  logarithm  belongs. 

In  the  common  tables,  the  indices  to  the  logarithms  of  the 
first  100  numbers  are  inserted.  But,  for  all  other  numbers, 
the  decimal  part  only  of  the  logarithm  is  given  ;  while  the 
index  is  left  to  be  supplied,  according  to  the  principles  in 
Arts.  8  and  11. 

25.  To  find  the  logarithm  of  any  number  between  1  and 
100: 

Look  for  the  proposed  number,  on  the  left ;  and  against  it, 
in  the  next  column,  will  be  the  logarithm,  with  its  index 
Thus, 

The  log.  of  18  is  1.25527.     The  log.  of  73  is  1.86332. 

26.  To  find  the,  logarithm  of  any  number  between  100  and 
1000  ;  or  of  any  number  consisting  of  not  more  than  three 
significant  figures,  with  ciphers  annexed. 

*  The  best  English  Tables  are  Button's  in  8vo.  and  Taylor's  in  4to. 
In  these,  the  logarithms  are  carried  to  seven  places  of  decimals,  and 
proportional  parts  are  placed  in  the  margin.  The  smaller  tables  are  nu- 
merous ;  and,  when  accurately  printed,  are  sufficient  for  common  cal- 
culations. 


THE    LOGARITHMIC  TABLES.  16 

In  the  smaller  tables,  the  three  first  figures  of  each  num- 
ber, are  generally  placed  in  the  left  hand  column  ;  and  the 
fourth  figure  is  placed  at  the  head  of  the  other  columns. 

Any  number,  therefore,  between  100  and  1000,  may  be 
found  on  the  left  hand  ;  and  directly  opposite,  in  the  next 
column,  is  the  decimal  part  of  its  logarithm.  To  this  the 
index  must  be  prefixed,  according  to  the  rule  in  Art.  8. 

The  log.  of  458  is  2.66087,     The  log.  of  935  is  2.97081, 
of  796      2.90091,  of  386      2.58659. 

If  there  are  ciphers  annexed  to  the  significant  figures,  the 
logarithm  may  be  found  in  a  similar  manner.  For,  by  Art. 
14,  the  decimal  part  of  the  logarithm  of  any  number  is  the 
same,  as  that  of  the  number  multiplied  into  10,  100,  &c. 
All  the  difference  will  be  in  the  index  ;  and  this  may  be  sup- 
plied by  the  same  general  rule. 

The  log.  of  4580  is  3.66087,  The  log.  of  326000  is  5.51322, 
of  79600      4.90091,  of  8010000       6.90363. 

27.  To  find  the  logarithm  of  any  number  consisting  of 
FOUR  figures,  either  with,  or  without,  ciphers  annexed. 

Look  for  the  three  first  figures,  on  the  left  hand,  and  for 
the  fourth  figure,  at  the  head  of  one  of  the  columns.  The 
logarithm  will  be  found,  opposite  the  three  first  figures,  and 
in  the  column  which,  at  the  head,  is  marked  with  the  fourth 
figure.* 

The  log.  of  6234  is  3.79477,    The  log.  of  788400  is  5.89398, 
of  5231      3.71858,  of  6281000     6.79803. 

28.  To  find  tlie  logarithm  of  a  number  containing  MORE 
than  FOUR  significant  figures. 

By  turning  to  the  tables,  it  will  be  seep,  that  if  the  differ- 
ences between  several  numbers  be  small,  in  comparison  with 
the  numbers  themselves  ;  the  differences  of  the  logarithms 

*  In  Taylor's,  Hutton's,  and  other  tables,  four  figures  are  placed  in 
the  left  hand  column,  and  the  fifth  at  the  top  of  the  page. 


10  THE    LOGARITHMIC    TABLES. 

will  be  nearly  proportioned  to  the  differences  of  the  numbers. 
Thus, 

The  log.  of  1000  is  3.00000,  Here  the  differences  in  the 
of  1001  3.00043,  numbers  are,  1,  2,  3,  4,  &c., 
of  1002  3.00087,  and  the  corresponding  dif- 
of  1003  3.00130,  ferences  in  the  logarithms, 
of  1004  3.00173,  <fec.  are  43,  87,  130,  173,  &c. 

Now  43  is  nearly  half  of  87,  one-third  of  130,  one-fourth 
of  173,  &c 

Upon  this  principle,  we  may  find  the  logarithm  of  a  num- 
ber which  is  between  two  other  numbers  whose  logarithms 
are  given  by  the  tables.  Thus,  the  logarithm  of  21716  is 
not  to  be  found  in  those  tables  which  give  the  numbers  to 
four  places  of  figures  only. 

But  by  the  table,  the  log.  of  21720  is  4.33686 
and  the  log.  of  21710  is  4.33666 

The  difference  of  the  two  numbers  is  10 ;  and  that  of  the 
logarithms  20. 

Also,  the  difference  between  21710,  and  the  proposed  num- 
ber 21716,  is  6. 

If,  then,  a  difference  of  10  in  the  numbers 
make  a  difference  of  20  in  the  logarithms : 

A  difference  of    6  in  the  numbers  will 
make  a  difference  of  12  in  the  logarithms. 

That  is,  10  :  20  : :  6  :  12. 

If,  therefore,  12  be  added  to  4.33666,  the  log.  of  21710 ; 
The  sum  will  be  4.33678,  the  log.  of  21716. 

We  have,  then,  this 

RULE. 

To  find  the  logarithm  of  a  number  consisting  of  more  than 
four  figures : 


THE    LOGARITHMIC    TABLES.  17 

Take  out  the  logarithm  of  two  numbers,  one  greater,  and 
the  other  less,  than  the  number  proposed :  Find  the  differ- 
ence of  the  two  numbers,  and  the  difference  of  their  loga- 
rithms :    Take  also  the  difference  between  the  least  of  the 
two  numbers,  and  the  proposed  number.     Then  say, 
As  the  difference  of  the  two  numbers, 
To  the  difference  of  their  logarithms  ; 
So  is  the  difference  between  the  least  of  the  two  nunv 

bers,  and  the  proposed  number, 
To  the  proportional  part  to  be  added  to 

the  least  of  the  two  logarithms. 

It  will  generally  be  expedient  to  make  \hefirstfourfigurc8, 
in  the  least  of  the  two  numbers,  the  same  as  in  the  proposed 
number,  substituting  ciphers,  for  the  remaining  figures ;  and 
to  make  the  greater  number  the  same  as  the  less,  with  the 
addition  of  a  unit  to  the  last  significant  figure.  Thus, 

For  36843,     take  36840,     and  36850, 
For  792674,  792600,          792700, 

For  6537825,         6537000,        6538000,  &c. 

The  first  term  of  the  proportion  will  then  be  10,  or  100, 
or  1000,  &c. 

Ex.  1.  Required  the  logarithm  of  362572. 

The  logarithm  of  362600  is  5.55943 
of  362500      5.55931 
The  differences  are        100,    and     12. 

Then  100  :  12  : :  72  :  8.64,  or  9  nearly. 
And  the  log.  5.55931+9=5.55940,  the  log.  required. 

Ex.  2.  The  log.  of  78264      is  4.89356 

3.  The  log.  of  143542    is  5.15698 

4.  The  log.  of  1129535  is  6.05290. 

By  a  little  practice,  such  a  facility  in  abridging  these  cal- 
culations may  be  acquired,  that  the  logarithms  may  be  taken 

2* 


18  THE   LOGARITHMIC    TABLES. 

out,  in  a  very  short  time.  When  great  accuracy  is  not  re- 
quired, it  will  be  easy  to  make  an  allowance  sufficiently  near, 
without  formally  stating  a  proportion.  In  the  larger  tables, 
the  proportional  parts  which  are  to  be  added  to  the  loga- 
rithms, are  already  prepared,  and  placed  in  the  margin. 

29.  To  find  the  logarithm  of  a  DECIMAL  FRACTION. 

The  logarithm  of  a  decimal  is  the  same  as  that  of  a  whole 
number,  excepting  the  index.  (Art.  14.)  To  find  then  the 
logarithm  of  a  decimal,  take  out  that  of  a  whole  number 
consisting  of  the  same  figures  ;  observing  to  make  the  neg- 
ative index  equal  to  the  distance  of  the  first  significant  figure 
of  the  fraction  from  the  place  of  units.  (Art.  11.) 

The  log.  of  0.07643,      is  "^88326,  or  8.88326,  (Art.  12.) 
of  0.00259,          |[41330,  or  7.41330, 
of  0.0006278,      4/79782,  or  6.79782. 

30.  To  find  the  logarithm  of  a  MIXED  decimal  number. 

Find  the  logarithm,  in  the  same  manner  as  if  all  the  fig- 
ures were  integers  ;  and  then  prefix  the  index  which  belongs 
to  the  integral  part,  according  to  Art.  8. 

The  logarithm  of  26.34  is  1.42062. 

The  index  here  is  1,  because  1  is  the  index  of  the  loga- 
rithm of  every  number  greater  than  10,  and  less  than  100. 
(Art.  7.) 

The  log.  of  2.36  is  0.37291,     The  log.  of  364.2  is  2.56134, 
of  27.8     1.44404,  of  69.42      1.84148. 

31.  To  find  the  logarithm  of  a  VULGAR  FRACTION. 

From  the  nature  of  a  vulgar  fraction,  the  numerator  may 
be  considered  as  a  dividend,  and  the  denominator  as  a  divi- 
sor ;  in  other  words,  the  value  of  the  fraction  is  equal  to 
the  quotient  of  the  numerator  divided  by  the  denominator. 
(Alg.  110.)  But  in  logarithms,  division  is  performed  by  sub- 
traction ;  that  is,  the  difference  of  the  logarithms  of  two  num- 
bers, is  the  logarithm  of  the  quotient  of  those  numbers. 


THE    LOGARITHMIC   TABLES.  19 

(Art.  1 .)  To  find  then  the  logarithm  of  a  vulgar  fraction, 
subtract  the  logarithm  of  the  denominator  from  that  of  the  nu- 
merator. The  difference  will  be  the  logarithm  of  the  frac- 
tion. Or  the  logarithm  may  be  found,  by  first  reducing  the 
vulgar  fraction  to  a  decimal.  If  the  numerator  is  less  than 
the  denominator,  the  index  of  the  logarithm  must  be  neg- 
ative, because  the  value  of  the  fraction  is  less  than  a  unit. 
(Art.  9.) 

Required  the  logarithm  of  ff . 

The  log.  of  the  numerator  is  1.53148 
of  the  denominator   1.93952 

of  the  fraction  J..591 96,  or  9.59196. 

The  logarithm  of  VVW  is  2^66362,  or  8.66362. 

of  T^TS     1T04376,  or  7.04376. 

32.  If  the  logarithm  of  a  mixed  number  is  required,  re- 
duce  it  to  an  improper  fraction,  and  then  proceed  as  before. 

The  logarithm  of  3$=^  is  0.57724. 


33.  To  find  the  NATURAL  NUMBER  belonging  to  any  loga- 
rithm. 

In  computing  by  logarithms,  it  is  necessary,  in  the  first 
place,  to  take  from  the  tables  the  logarithms  of  the  numbers 
which  enter  into  the  calculation  ;  and,  on  the  other  hand,  at 
the  close  of  the  operation,  to  find  the  number  belonging  to 
the  logarithm  obtained  in  the  result.  This  is  evidently  done 
by  reversing  the  methods  in  the  preceding  articles. 

Where  great  accuracy  is  not  required,  look  in  the  tables 
for  the  logarithm  which  is  nearest  to  the  given  one  ;  and  di- 
rectly opposite  on  the  left  hand,  will  be  found  the  three  first 
figures,  and  at  the  top,  over  the  logarithm,  the  fourth  figure 
of  the  number  required.  This  number,  by  pointing  off  dec- 


20  THE   LOGARITHMIC   TABLES. 

imals,  or  by  adding  ciphers,  if  necessary,  must  be  made  to 
correspond  with  the  index  of  the  given  logarithm,  according 
to  Arts.  8  and  11. 

The  natural  number  belonging 

to  3.86493  is  7327,         to  1.62572  is  42.24, 
to  2.90141      796.9,        toT.89115       0.07783. 

In  the  last  example,  the  index  requires  that  the  first  signi- 
ficant figure  should  be  in  the  second  place  from  units,  and 
therefore  a  cipher  must  be  prefixed.  In  other  instances,  it 
is  necessary  to  annex  ciphers  on  the  right,  so  as  to  make 
the  number  of  figures  exceed  the  index  by  1. 

The  natural  number  belonging 

to  6.71567  is  5196000,  to  ^65677  is  0.004537, 
to  4.67062  46840,   to   4~59802       0.0003963. 

34.  When  great  accuracy  is  required,  and  the  given  loga- 
rithm is  not  exactly,  or  very  nearly,  found  in  the  tables,  it 
will  be  necessary  to  reverse  the  rule  in  Art.  28. 

Take  from  the  tables  two  logarithms,  one  the  next  greater, 
the  other  the  next  less  than  the  given    logarithm.     Find 
the  difference  of  the  two  logarithms,  and  the  difference  of 
their  natural  numbers  ;  also  the  difference  between  the  least 
of  the  two  logarithms,  and  the  given  logarithm.     Then  say, 
As  the  difference  of  the  two  logarithms, 
To  the  difference  of  their  numbers ; 
So  is  the  difference  between  the  given 

logarithm  and  the  least  of  the  other  two, 
To  the  proportional  part  to  be  added  to 
the  least  of  the  two  numbers. 

Required  the  number  belonging  to  the  logarithm  2.67325. 
Next  great. log.2.67330.  Its  numb.  471.3.  Given  log.  2. 67325. 
Next  less  2.67321.  Its  numb.  471.2.  Next  less  2.67321. 
Differences  T  0.1  ~~4 


THE    LOGARITHMIC    TABLES.  21 

Then,  9  :  0.1  :  :  4  :  0.044,  which  is  to  be  added 

to  the  number  471.2 
The  number  required  is       471.244. 

The  natural  number  belonging 

to  4.37627  is  23783.45,     to  JL73698  is  54.57357, 
to  3.69479      4952.08,       to  T09214      0.123635. 


35.  Correction  of  the  Tables. — The  tables  of  logarithms 
have  been  so  carefully  and  so  repeatedly  calculated,  by  the 
ablest  computers,  that  there  is  no  room  left  to  question  their 
general  correctness.  They  are  not,  however,  exempt  from 
the  common  imperfections  of  the  press.  But  an  error  of  this 
kind  is  easily  corrected,  by  comparing  the  logarithm  with 
any  two  others  to  whose  sum  or  difference  it  ought  to  be 
equal.  (Art.  1.) 

Thus  48=24X2=16X3=12X4=8X6.  Therefore,  the 
logarithm  of  48  is  equal  to  the  sum  of  the  logarithms  of  24 
and  2,  of  16  and  3,  &c. 

And,  3=f =Y=-¥L=-Vl=-y-,  &c.  Therefore,  the  loga- 
rithm of  3  is  equal  to  the  diference  of  the  logarithms  of  6 
and  2,  of  12  and  4,  &c. 


22  MULTIPLICATION    BY    LOGARITHMS. 


SECTION  III. 


R£ETHODS    OF    CALCULATING    BY    LOGARITHMS. 

ART.  36.  The  arithmetical  operations  for  which  logarithms 
were  originally  contrived,  and  on  which  their  great  utility 
depends,  are  chiefly  multiplication,  division,  involution,  evo- 
lution, and  finding  the  term  required  in  single  and  compound 
proportion.  The  principle  on  which  all  these  calculations 
are  conducted,  is  this  : 

If  the  logarithms  of  two  numbers  be  added,  the  SUM  will  be 
the  logarithm  of  the  PRODUCT  of  the  numbers  ;  and, 

If  the  logarithm  of  one  number  be  subtracted  from  that  of 
another,  the  DIFFERENCE  will  be  the  logarithm  of  the  QUOTIENT 
of  one  of  the  numbers  divided  by  the  other. 

In  proof  of  this,  we  have  only  to  call  to  mind,  that  loga- 
rithms are  the  EXPONENTS  of  a  series  of  powers  and  roots. 
(Arts.  2,  5.)  And  it  has  been  shown,  that  powers  and  roots 
are  multiplied  by  adding  their  exponents ;  and  divided,  bv 
subtracting  their  exponents.  (Alg.  189,  193,  232,  239.) 

MULTIPLICATION    BY    LOGARITHMS. 

37.  ADD  THE  LOGARITHMS  OF  THE  FACTORS  '.  THE  SUM 
WILL  BE  THE  LOGARITHM  OF  THE  PRODUCT. 

In  making  the  addition,  1  is  to  be  carried  for  every  10, 
from  the  decimal  part  of  the  logarithm,  to  the  index.  (Art.  7.) 

Numbers.  Logarithms.  Numbers.    Logarithms. 

Mult.  36.2  (Art.  30.)  1.55871         Mult     640       2.80618 

Into     7.84  0.89432         Into  2.316       0.36474 

Prod.     283.8  "2745303       Prod.   1482       3.17092 

The   logarithms  of  the   two  factors  are  taken  from  the 


MULTIPLICATION    BY    LOGARITHMS.  23 

tables.     The  product  is  obtained,  by  finding,  in  the  tables, 
the  natural  number  belonging  to  the  sum.  (Art.  33.) 

Mult.  89.24         1.95056  Mult.    134.         2.12710 

Into     3.687         0.56667  Into      25.6         1.40824 

Prod.  329.  2.51723  Prod.  3430          3.53534 

38.  When  any  or  all  of  the  indices  of  the  logarithms  are 
negative,  they  are  to  be  added  according  to  the  rules  for  the 
addition  of  positive  and  negative  quantities  in  algebra.  But 
it  must  be  kept  in  mind,  that  the  decimal  part  of  the  loga- 
rithm is  positive.  (Art.  10.)  Therefore,  that  which  is  car- 
ried from  the  decimal  part  to  the  index,  must  be  considered 
positive^also. 

Mult.   62.84        J..79824  Mult.  0.0294        ¥.46835 

Into     0.682        T.83378  Into     0.8372         T.92283 

Prod.  42.86         1.63202  Prod.  0.0246         ^39118 

In  each  of  these  examples,  +1  is  to  be  carried  from  the 
decimal  part  of  the  logarithm.  This,  added  to  — 1,  the 
lower  index,  makes  it  0 ;  so  that  there  is  nothing  to  be 
added  to  the  upper  index. 

If  any  perplexity  is  occasioned,  by  the  addition  of  positive 
and  negative  quantities,  it  may  be  avoided,  by  borrowing  10 
to  the  index.  (Art.  12.) 

Mult.  .62.84         1.79824  Mult.  0.0294         8.46835 

Into     0.682          9.83378  Into     0.8372          9.92283 

Prod.  42.86         1.63202  Prod.  0.0246         8.39118 

Here  10  is  added  to  the  negative  indices,  and  afterwards 
rejected  from  the  index  of  the  sum  of  the  logarithms. 

Multiply        26.83  1.42862  1.42862 

Into  0.00069  4~83885    or    6.83885 

Product    0.01851          £726747          8.26747 


24  MULTIPLICATION    BY   LOGARITHMS. 

Here  +1  carried  to  — 4  makes  it  — 3,  which  added  to  the 
upper  index  +1,  gives  — 2  for  the  index  of  the  sum. 

Multiply      .00845          1T92686    or    7.92686 
Into  1068.  3.02857  3.02857 

Product      9.0246  0.95543  0.95543 

The  product  of  0.0362     into  25.38       is   0.9188 
of  0.00467  into  348.1       is    1.626 
of  0.0861     into  0.00843  is   0.0007258 

39.  Any  number  of  factors  may  be  multiplied  together,  by 
adding  their  logarithms.  If  there  are  several  positive,  and 
several  negative  indices,  these  are  to  be  reduced  to  one,  as  in 
algebra,  by  taking  the  difference  between  the  sum  of  those 
which  are  negative,  and  the  sum  of  those  which  are  positive, 
increased  by  what  is  carried  from  the  decimal  part  of  the 
logarithms.  (Alg.  53.) 


Multiply      6832 
Into           0.00863 

3.83455 
3.93601 

3.83455 
or    7.93601 

And          0.651 

1.81358 

9.81358 

And          0.0231 

1.36361 

or    8.36361 

And          62.87 

1.79844 

1.79844 

Prod.        55.74 

1.74619 

1.74619 

Ex.  2.  The  prod,  of  36.4x7.82X68.91X0.3846  is  7544. 

3.  The  prod,  of  0.00629X2.647X0.082X278.8X0.00063 
is  0.0002398. 

40.  Negative  quantities  are  multiplied,  by  means  of  loga- 
rithms, in  the  same  manner  as  those  which  are  positive.  (Art. 
16.)  But,  after  the  operation  is  ended,  the  proper  sign  must 
be  applied  to  the  natural  number  expressing  the  product, 
according  to  the  rules  for  the  multiplication  of  positive  and 
negative  quantities  in  algebra.  The  negative  index  of  a  loga- 
rithm, must  not  be  confounded  with  the  sign  which  denotes 
that  the  natural  number  is  negative.  That  which  the  index 


DIVISION    BY    LOGARITHMS.  25 

of  the  logarithm  is  intended  to  show,  is  not  whether  the 
natural  number  is  positive  or  negative,  but  whether  it  is 
greater  or  less  than  a  unit.  (Art.  16.) 

Mult.  +36.42  1.56134  Mult.  —2.681  0.42830 
Into  — 67.31  1.82808  Into  +37.24  1.57101 
Prod.  —2451  3.38942  Prod.  —99.84  1.99931 

In  these  examples,  the  logarithms  are  taken  from  the 
tables,  and  added,  in  the  same  manner,  as  if  both  factors 
were  positive.  But  after  the  product  is  found,  the  negative 
sign  is  prefixed  to  it,  because  +  is  multiplied  into  — .  (Alg. 
82.) 

Mult.  0.263  T.41996         Mult.  0.065  2^81291 

Into  0.00894  T.95134  Into  0.693  T84073 
Prod.  0.002351  lf.37130  Prod.  0.04504  1^65364 

Here  the  indices  of  the  logarithms  are  negative,  but  the 
product  is  positive,  because  the  factors  are  both  positive. 

Mult.  — 62.59  _L79650  Mult  — 68.3  1.83442 
Into  — 0.00863  "¥.93601  Into  — 0.0096  ~3.98227 
Prod.  +0.5402  T.73251  Prod.  +0.6557  T.81669 


DIVISION    BY    LOGARITHMS. 

41.  FROM  THE   LOGARITHM  OF  THE  DIVIDEND,    SUB- 
TRACT    THE     LOGARITHM     OF     THE     DIVISOR;    THE    DIF- 
FERENCE     WILL      BE      THE      LOGARITHM      OF     THE      QUO- 

1'IENT.  (Art.  36.) 

Logarithms. 
2.95245 
0.99330 
1.95915 

42.  The  decimal  part  of  the  logarithm  may  be  subtracted 

8 


Numbers. 

Divide   6238 

Logarithms. 

3.79505 

Numbers. 

Divide    896.3 

By         2982 

3.47451 

By          9.847 

Cluot.    2.092 

0.32054 

Quot.     91.02 

26 


DIVISION    BY   LOGARITHMS. 


as  in  common  arithmetic.  But  for  the  indices,  -when  either 
<of  them  is  negative,  or  the  lower  one  is  greater  than  the 
upper  one,  it  will  be  necessary  to  make  use  of  the  general 
rule  for  subtraction  in  algebra ;  that  is,  to  change  the  signs  of 
the  subtrahend,  and  then  proceed  as  in  addition.  (Alg.  60.) 
When  1  is  carried  from  the  decimal  part,  this  is  to  be  con- 
sidered affirmative,  and  applied  to  the  index,  before  the  sign 
is  changed. 

Divide  0.8697        T.93937   or   9.93937 
By         98.65  1.99410         1.99410 

Quot.     0.008816     3^94527          7.94527 

In  this  example,  the  upper  logarithm  being  less  than  the 
lower  one,  it  is  necessary  to  borrow  10,  as  in  other  cases  of 
subtraction ;  and  therefore  to  carry  one  to  the  lower  indeT, 
which  then  becomes  +2.  This  changed  to  — 2,  and  added 
to  — 1  above  it,  makes  the  index  of  the  difference  of  the 
logarithms  — 3. 

Divide  29.76  1.47363         1.47363 

By  6254  3.79616          3.79616 

Quot.      0.00476      3~67747   or  7.67747 

Here,  1  carried  to  the  lower  index,  makes  it  +4.  This 
changed  to  — 4,  and  added  to  1  above  it,  gives  — 3  for  the 
index  of  the  difference  of  the  logarithms. 

Divide  6.832  -  _0.8S455  Divide  0.00634  "£80209 
By  .0362  12.55871  By  62.18  1.79365 

Quot.    188.73        2.27584         Quot.       0.000102      Z00844 

The  quotient  of  0.0985  divided  by  0.007241,  is  13.6 
The  quotient  of  0.0621  divided  by  3.68,  is  0.01687 

43.  To  divide  negative  quantities,  proceed  in  the  same 
manner  as  if  they  were  positive,  (Art.  40.)  and  prefix  to  the 


INVOLUTION   BY   LOGARITHMS.  27 

quotient,  the  sign  which  is  required  by  the  rules  for  division 
in  algebra. 

Divide  +3642  3.56134  Divide  —0.657  T81757 
By  — 23.68  1.37438  By  +0.0793  2.89927 
Quot.  —153.8  2.18696  Quot.  —8.285  0.91830 

In  these  examples,  the  sign  of  the  divisor  being  different 
from  that  of  the  dividend,  the  sign  of  the  quotient  must  be 
negative.  (Alg.  100.) 

Divide  — 0.364  T.56110  Divide  — 68.5  1.83569 
By  —2.56  0.40824  By  +0.094  ¥.97313 
Quot.  +0.1422  Tl5286  Quot.  — 728.7  2.86256 


INVOLUTION    BY   LOGARITHMS. 

44.  Involving  a  quantity  is  multiplying  it  into  itself.  By 
means  of  logarithms,  multiplication  is  performed  by  addition. 
If,  then,  the  logarithm  of  any  quantity  be  added  to  itself,  the 
logarithm  of  a  power  of  that  quantity  will  be  obtained.  But 
adding  a  logarithm,  or  any  other  quantity,  to  itself,  is  mul- 
tiplication. The  involution  of  quantities,  by  means  of  loga- 
rithms, is  therefore  performed,  by  multiplying  the  logarithms. 

Thus  the  logarithm 

of  100  is  2 

of  100X100,  that  is,  of  100*  is  2+2  =2X2 

of  100X100X100,  To08is2+2+2  =2X3 

of  100X100X100X100,100*  is  2+2+2+2     =2X4 


On  the  same  principle,  the  logarithm  of  100n  is 
And  the  logarithm  of  xn,  is  (log.  x)Xn.     Hence, 

45.    To   involve  a  quantity  by  logarithms,  MULTIPLY 

THE   LOGARITHM    OF   THE    QUANTITY,    BY   THE    INDEX    OF  THE 
POWER    REQUIRED. 


28  INVOLUTION   BY   LOGARITHMS. 

The  reason  of  the  rule  is  also  evident,  from  the  considera- 
tion, that  logarithms  are  the  exponents  of  powers  and  roots, 
and  a  power  or  root  is  involved,  by  multiplying  its  index 
into  the  index  of  the  power  required.  (Alg.  170,  242.) 

Ex.  1.  What  is  the  cube  of  6.296  ? 

Root     6.296,            its  log.  0.79906 

Index  of  the  power  3 

Power  249.6  2.39718 

2.  Required  the  4th  power  of     21.32 
Root     21.32  log.          1.32879 

Index  4 

Power  206614  5.31516 

3.  Required  the  6th  power  of     1.689 
Root     1.689  log.          0.22763 

Index  6 

Power  23.215  1.36578 

4.  Required  the  144th  power  of     1.003 
Root     1.003  log.          0.00130 

Index  144 

Power  1.539  0.18720 

46.  It  must  be  observed,  as  in  the  case  of  multiplication, 
(Art.  38.)  that  what  is  carried  from  the  decimal  part  of  the 
logarithm  is  positive,  whether  the  index  itself  is  positive  or 
negative.  Or,  if  10  be  added  to  a  negative  index,  to  render 
it  positive,  (Art,  12.)  this  will  be  multiplied,  as  well  as  the 
other  figures,  so  that  the  logarithm  of  the  square,  will  be  20 
too  great ;  of  the  cube,  30  too  great,  &c. 

Ex.  1.  Required  the  cube  of          0.0649 
Root     0.0649         log.  2^81224  or  8.81224 

Index  3  3 

Power  0.0002733  4~43672         6.43672 


EVOLUTION    BY   LOGARITHMS.  29 

2.  Required  the  4th  power  of       0.1234 

Root     0.1234         log.                T09132  or  9.09132 

Index  4        4 

Power  0.0002319                        T.36528  6.36528 

8.  Required  the  6th  power  of       0.9977 

Root     0.9977         log.                T.99900  or  9.99900 

Index                         6  6 

Power  0.9863                               T.99400  9.99400 

4.  Required  the  cube  of  0.08762 

Root     0.08762       log.                1F.94260  or  8.94260 

Index  8        3 

Power  0.0006727                         T82780  6.82780 

5.  The  7th  power  of  0.9061  is  0.5015. 

6.  The  5th  power  of  0.9344  is  0.7123. 


EVOLUTION   BT   LOGARITHMS. 

« 

47.  Evolution  is  the  opposite  of  involution.  Therefore, 
as  quantities  are  involved,  by  the  multiplication  of  logarithms, 
roots  are  extracted  by  the  division  of  logarithms  ;  that  is, 

To  extract  the  root  of  a  quantity  by  logarithms,  DIVIDE 

THE     LOGARITHM     OF     THE     QUANTITY,    BY     THE     NUMBER    EX- 
PRESSING  THE    ROOT    REQUIRED. 

The  reason  of  the  rule  is  evident  also,  from  the  fact,  that 
logarithms  are  the  exponents  of  powers  and  roots,  and  evo- 
lution is  performed,  by  dividing  the  exponent,  by  the  number 
expressing  the  root  required.  (Alg.  210.) 

1.  Required  the  square  root  of  648.3 

Numbers.  Logarithms. 

Power   648.3  2)2.81178 

Root      25.46  1.40589 

3* 


30  EVOLUTION   BY    LOGARITHMS. 

2.  Required  the  cube  root  of  897.1 

Power   897.1  3)2.95284 

Root      9.645  0.98428 

In  the  first  of  these  examples,  the  logarithm  of  the  give* 
number  is  divided  by  2  ;  in  the  other,  by  3. 

8.  Required  the  10th  root  of  6948. 

Power    6948  10)3.84186 

Root      2.422  0.38418 

4.  Required  the  lOOdth  root  of  983. 

Power      983         100)2.99255 
Root      1.071  0.02992 

The  division  is  performed  here,  as  in  other  cases  of  deci- 
mals, by  removing  the  decimal  point  to  the  left. 

5.  What  is  the  ten  thousandth  root  of  49680000  ? 

Power  49680000  10000)7.69618 

Root     1.00179  0.00077 

We  have,  here,  an  example  of  the  great  rapidity  with 
which  arithmetical  operations  are  performed  by  logarithms. 

48.  If  the  index  of  the  logarithm  is  negative,  and  is  not 
divisible  by  the  given  divisor,  without  a  remainder,  a  diffi- 
culty will  occur,  unless  the  index  be  altered. 

Suppose  the  cube  root  of  0.0000892  is  required.  The 
logarithm  of  this  is  5.95036.  If  we  divide  the  index  by  3, 
the  quotient  will  be  — 1,  with  — 2  remainder.  This  remain- 
der, if  it  were  positive,  might,  as  in  other  cases  of  division, 
be  prefixed  to  the  next  figure.  But  the  remainder  is  nega- 
tive, while  the  decimal  part  of  the  logarithm  is  positive  ;  so 
that,  when  the  former  is  prefixed  to  the  latter,  it  will  make 
neither  +2.9  nor — 2.9,  but — 2 +  .9.  This  embarrassing 
intermixture  of  positives  and  negatives  may  be  avoided,  by 
adding  to  the  index  another  negative  number,  to  make  it  ex- 


EVOLUTION    BY    LOGARITHMS.  31 

actly  divisible  by  the  divisor.  Thus,  if  to  the  index  — 5 
there  be  added  — 1,  the  sum  — 6  will  be  divisible  by  3.  But 
this  addition  of  a  negative  number  must  be  compensated,  by 
the  addition  of  an  equal  positive  number,  which  may  be  pre- 
fixed to  the  decimal  part  of  the  logarithm.  The  division 
may  then  be  continued,  without  difficulty,  through  the 
whole. 

Thus,  if  the  logarithm  ~5~.95036  be  altered  to ~6~+ 1.95036 
it  may  be  divided  by  3,  and  the  quotient  will  be  2.65012. 
We  have  then  this  rule, 

49.  Add  to  the  index,  if  necessary,  such  a  negative  num- 
b&  a&  mil  make  it  exactly  divisible  by  the  divisor,  and  prefix 
an  equal  positive  number  to  the  decimal  part  of  the  logarithm. 

1.  Required  the  5th  root  of  0.009642 

Power  0.009642     log.  3^98417 

or    "6+2.98417 

Root         0.3952  1.59683 

2.  Required  the  7th  root  of  0.0004935. 

Power  0.0004935  log.  1L69329 

or  7)7+3.69329 

Root  0.337  T.52761 

50.  If,  for  the   sake  of  performing  the  division   conven- 
iently, the  negative  index  be  rendered  positive,  it  will  be  ex- 
pedient to  borrow  as  many  tens,  as  there  are  units  in  the 
number  denoting  the  root. 

What  is  the  fourth  root  of  0.03698  ? 

Power  0.03698         4)^56797  or  4)38.56797 
Root      0.4385  T.64199  9.64199 

Here  the  index,  by  borrowing,  is  made  40  too  great,  that 
is,  +38  instead  of  — 2.  When,  therefore,  it  is  divided  by  4, 
it  is  still  10  too  great,  +9  instead  of  — 1. 


32  PROPORTION   BY   LOGARITHMS. 

What  is  the  5th  root  of  0.008926  ? 

Power  0.008926       5)3^95066  or  5)47.95066 
Hoot      0.38916  T.59013  9.59013 

61.  A  power  of  a  root  may  be  found  by  first  multiplying 
the  logarithm  of  the  given  quantity  into  the  index  of  the 
power,  (Art.  45.)  and  then  dividing  the  product  by  the 
number  expressing  the  root.  (Art.  47.) 

1.  What  is  the  value  of  (53)^,  that  is,  the  6th  power  of 
the  7th  root  of  53  ? 

Given  number  53      log.       1.72428 
Multiplying  by  6 

Dividing  by  7)10.34568 

Power  required  30.06          1.47795 

2.  What  is  the  8th  power  of  the  9th  root  of  654  ? 

PROPORTION    BY   LOGARITHMS. 

52.  In  a  proportion,  when  three  terms  are  given,  the 
fourth  is  found  in  common  arithmetic,  by  multiplying  to- 
gether the  second  and  third,  and  dividing  by  the  first.  But 
when  logarithms  are  used,  addition  takes  the  place  of  mul- 
tiplication, and  subtraction,  of  division. 

To  find,  then,  by  logarithms,  the  fourth  term  in  a  propor- 
tion, ADD  THE  LOGARITHMS  OF  THE  SECOND  AND  THIRD 

TERMS,  AND  from  the  sum  SUBTRACT  THE  LOGARITHM  OF 
THE  FIRST  TERM.  The  remainder  will  be  the  logarithm  of 
the  term  required. 

Ex.  1.  Find  a  fourth  proportional  to  7964,  378,  and  27960. 

Numbers.  Logarithms. 

Second  term  378  2.57749 

Third  term          27960  4.44654 

7.02403 

First  term  7964  3.90113 

Fourth  term          1327  3.12290 


ARITHMETICAL  COMPLEMENT.  $3 

2.  Find  a  4th  proportional  to  768,  381,  and  9780. 
Second  term  381  2.58092 

Third  term  9780  3.99034 

6.57126 
First  term  768  2.88536 

Fourth  term          4852  3.68590 

ARITHMETICAL    COMPLEMENT. 

53.  When  one  number  is  to  be  subtracted  from  another, 
it  is  often  convenient,  first  to  subtract  it  from  10,  then  to 
add  the  difference  to  the  other  number,  and  afterwards  to  re- 
ject the  10. 

Thus,  instead  of  a — b,  we  may  put  10 — b-\-a — 10. 

In  the  first  of  these  expressions,  b  is  subtracted  from  a. 
In  the  other,  b  is  subtracted  from  10,  the  difference  is  added 
to  a,  and  10  is  afterwards  taken  from  the  sum.  The  two  ex- 
pressions are  equivalent,  because  they  consist  of  the  same 
terms,  with  the  addition,  in  one  of  them,  of  10 — 10=0.  The 
alteration  is,  in  fact,  nothing  more  than  borrowing  10,  for  the 
sake  of  convenience,  and  then  rejecting  it  in  the  result. 

Instead  of  10,  we  may  borrow,  as  occasion  requires,  100, 
1000,  &c. 

Thus,  a— 6=100— 6+a— 100=1000— b+a— 1000,  &c. 

54.  The  DIFFERENCE   between  a  given  number  and  10,  or 
100,  or  1000,  &c.y  is  called  the  ARITHMETICAL  COMPLEMENT 
of  that  number. 

The  arithmetical  complement  of  a  number  consisting  of 
one  integral  figure,  either  with  or  without  decimals,  is  found, 
by  subtracting  the  number  from  10.  If  there  are  two  in- 
tegral figures,  they  are  subtracted  from  100 ;  if  three,  from 
1000,  &c. 

Thus,  the  arithmetical  compl't  of  3.46  is  10—3.46=6.54 

of  34.6  is  100—34.6=65.4 
of  346.  is  1000— 346.=654.  &c. 


84  ARITHMETICAL    COMPLEMENT. 

According  to  the  rule  for  subtraction  in  arithmetic,  any 
number  is  subtracted  from  10,  100,  1000,  &c.  by  beginning 
on  the  right  hand,  and  taking  each  figure  from  10,  after  in- 
creasing all  except  the  first,  by  carrying  1. 

Thus,  if  from  10.00000 

We  subtract  7.63125 

The  difference,  or  arith'l  compl't  is  2.36875,  which  is 
obtained  by  taking  5  from  10,  3  from  10,  2  from  10,  4  from 
10,  7  from  10,  and  8  from  10.  But,  instead  of  taking  each 
figure,  increased  by  1  from  10 ;  we  may  take  it  without 
being  increased,  from  9. 

Thus,  2  from  9  is  the  same  as  3  from  10, 

3  from  9      the  same  as  4  from  10,  <fec.     Hence, 

55.  To  obtain  the  ARITHMETICAL  COMPLEMENT  of  a  num- 
ber, subtract  the  right  hand  significant  figure  from  10,  and 
each  of  the  other  figures  from  9.  If,  however,  there  are 
ciphers  on  the  right  hand  of  all  the  significant  figures,  they 
are  to  be  set  down  without  alteration. 

In  taking  the  arithmetical  complement  of  a  logarithm,  if 
the  index  is  negative,  it  must  be  added  to  9  ;  for  adding  a 
negative  quantity  is  the  same  as  subtracting  a  positive  one. 
(Alg.  81.)  The  difference  between  — 3  and  +9,  is  not  6, 
but  12. 

The  arithmetical  complement 

of  6.24897  is  3.75103  of  "2^70649  is  11.29351 
of  2.98643  7.01357  of  3.64200  6.35800 
of  0.62430  9.37570  of  9.35001  0.64999 

66.  The  principal  use  of  the  arithmetical  complement,  is 
in  working  proportions  by  logarithms ;  where  some  of  the 
terms  are  to  be  added,  and  one  or  more  to  be  subtracted. 
In  the  Rule  of  Three  or  simple  proportion,  two  terms  are  to 
be  added,  and  from  the  sum,  the  first  term  is  to  be  sub- 
tracted. But  if,  instead  of  the  logarithm  of  the  first  term, 


COMPOUND    PROPORTION.  35 

we  substitute  its  arithmetical  complement,  this  may  be  added 
to  the  sum  of  the  other  two,  or  more  simply  all  three  may- 
be added  together,  by  one  operation.  After  the  index  is 
diminished  by  10,  the  result  will  be  the  same  as  by  the  com- 
mon method.  For  subtracting  a  number  is  the  same,  as  adding 
its  arithmetical  complement,  and  then  rejecting  10,  100,  or 
1000,  from  the  sum.  (Art.  53.) 

It  will  generally  be  expedient,  to  place  the  terms  in  the 
same  order,  hi  which  they  are  arranged  in  the  statement  of 
the  proportion, 

1.  As      6273    a.  c.  6,20252     2.  As      253  a.  c.  7.59688 
Is  to  769.4  2.88615  Is  to  672.5         2.82769 

So  is  37*61  1.57530          So  is  497  2.69636 

To      4.613  0.66397   *       To       1321.1       3.12093 

3.  As      46.34  a.  c.  8.33404  4.  As      9.85  a.  c.  9.00656 

Is  to  892.1           2.95041  Is  to  643             2.80821 

So  is  7.638           0.88298  So  is  76.3           1.88252 

To      147              2.16743  To     4981          3.69729 


COMPOUND    PROPORTION, 

57.  In  compound,  as  in  single  proportion,  the  term  re- 
quired may  be  found  by  logarithms,  if  we  substitute  additioD 
for  multiplication,  and  subtraction  for  division. 

Ex.  1.  If  the  interest  of  $365,  for  3  years  and  9  months, 
be  $82.13  ;  what  will  be  the  interest  of  $8940,  for  2  years 
and  6  months  ? 

In  common  arithmetic,  the  statement  of  the  question  is 
made  in  this  manner. 

365     dollars  (€940  dollars  >   .  . 


)    . 

f 


3.75    years      f       (     2.5  years 


06  COMPOUND    PROPORTION. 

And  the  method  of  calculation  is,  to  divide  the  product  of 
the  third,  fourth,  and  fifth  terms,  by  the  product  of  the  first 
two.*  This,  if  logarithms  are  used,  will  be  to  subtract  the 
sum  of  the  logarithms  of  the  first  two  terms,  from  the  sum 
of  the  logarithms  of  the  other  three. 

(     365  log.      2.56229 
First  two  terms       j    ^  0.57403 

Sum  of  the  logarithms  3.13632 

8940  3.95134 


Third  and  fourth  terms 

Fifth  term  82.13  1.91450 

Sum  of  the  logs,  of  the  3rd,  4th,  and  5th,  6.26378 

Do.  1st  and  2nd,  3.13632 

Term  required  1341  3.12746 

58.  The  calculation  will  be  more  simple,  if,  instead  of 
subtracting  the  logarithms  of  the  first  two  terms,  we  odd 
their  arithmetical  complements.  But,  it  must  be  observed, 
that  each  arithmetical  complement  increases  the  index  of  the 
logarithm  by  10.  If  the  arithmetical  complement  be  intro- 
duced into  two  of  the  terms,  the  index  of  the  sum  of  the 
logarithms  will  be  20  too  great  ;  if  it  be  in  three  terms,  the 
index  will  be  30  too  great,  &c. 

_.    A  A  I  365   a.  c.  7.43771 

First  two  terms       j  3.75  a.  c.  9<42597 

t  8940          3.95134 
Third  and  fourth  terms        ^          Q 


Fifth  term  82.13  1.91450 

Term  required         1341        23.12746 

The  result  is  the  same  as  before,  except  that  the  index  of 
the  logarithm  is  20  too  great. 

•>  See  Arithmetic. 


COMPOUND    INTEREST.  3 

Ex.  2.  If  the  wages  of  53  men  for  42  days  be  2200  dol- 
lars ;  what  will  be  the  wages  of  87  men  for  34  days  ? 

53  men     )        (    87  men     ) 

42days    |   :     j    34  days    {    : :  22°°  ' 

(     53.  a.  c.  8.27572 
First  two  terms       j     42.  a.  c.  8.37675 

.  (87  1.93952 

Third  and  fourth  terms       i     „ .  153148 

Fifth  term        2200  3.34242 

Term  required      2923.5  3.46589 

59.  In  the  same  manner,  if  the  product  of  any  number  of 
quantities,  is  to  be  divided,  by  the  product  of  several  others  ; 
we  may  add  together  the  logarithms  of  the  quantities  to  be 
divided,  and  the  arithmetical  complements  of  the  logarithms 
of  the  divisors. 

Ex.  If  29.67X346.2  be  divided  by  69.24x7.862X497  ; 
what  will  be  the  quotient  ? 

(    29.67  1.47232 

Numbers  to  be  divided     j    346  2  2 53933 

(    69.24  a.c.  8.15964 

Divisors  !    7.862  a.  c.  9.10447 

(       497  a.c.  7.30364 

Quotient  0.03797  8.5794 

In  this  way,  the  calculations  in  Conjoined  Proportion  may- 
be expeditiously  performed. 


COMPOUND    INTEREST. 

60.  In  calculating  compound  interest,  the  amount  for  the 
first  year,  is  made  the  principal  for  the  second  year ;  the 
amount  for  the  second  year,  the  principal  for  the  third 

4 


38  COMPOUND   INTEREST. 

year,  &c.  Now  the  amount  at  the  end  of  each  year,  must  be 
proportioned  to  the  principal  at  the  beginning  of  the  year.  If 
the  principal  for  the  first  year  be  1  dollar,  and  if  the  amount 
of  1  dollar  for  1  year=a;  then,  (Alg.  341.) 

',  :  aa=the  amount  for  the  2d  year,  or  the  prin- 
cipal for  the  3d ; 

;9  :  a9=the  amount  for  the  third  year,  or  the 
principal  for  the  4th ; 

,3  :  a*=the  amount  for  the  4th  year,  or  the  prin- 
cipal for  the  5th. 

That  is,  the  amount  of  1  dollar  for  any  number  of  years 
is  obtained  by  finding  the  amount  for  1  year,  and  involving 
this  to  a  power  whose  index  is  equal  to  the  number  of  years. 
And  the  amount  of  any  other  principal,  for  the  given  time, 
is  found  by  multiplying  the  amount  of  1  dollar,  into  the  num- 
ber of  dollars,  or  the  fractional  part  of  a  dollar. 

If  logarithms  are  used,  the  multiplication  required  here 
may  be  performed  by  addition  ;  and  the  involution  by  mul- 
tiplication. (Art.  45.)  Hence, 

61.  To  calculate  Compound  Interest,  Find  the  amount  of 
1  dollar  for  I  year;  multiply  its  logarithm  by  the  number  of 
years;  and  to  the  product,  add  the  logarithm  of  the  principal. 
The  sum  will  be  the  logarithm  of  the  amount  for  the  given 
time.  From  the  amount  subtract  the  principal,  and  the  re- 
mainder will  be  the  interest. 

If  the  interest  becomes  due  'half  yearly  or  quarterly  ;  find 
the  amount  of  one  dollar,  for  the  half  year  or  quarter,  and 
multiply  the  logarithm  by  the  number  of  half  years  or  quar- 
ters in  the  given  time. 

If  P=the  principal, 

a=the  amount  of  1  dollar  for  1  year, 
w=any  number  of  years,  and 
A=the  amount  of  the  given  principal  for  n  years ;  then. 


COMPOUND    INTEREST.  39 

Taking  the  logarithms  of  both  sides  of  the  equation,  and 
reducing  it,  so  as  to  give  the  value  of  each  of  the  four  quan- 
tities, in  terms  of  the  others,  we  have 

1.  Log.  A=nX  log.  a+  log.  P. 

2.  Log.  P=log.  A — nX  log.  a. 

log.  A— log.  P. 

3.  Log.  a—— 


4.  n= 


n 

log.  A— log.  P. 
log  a. 


Any  three  of  these  quantities  being  given,  the  fourth  may 
be  found. 

Ex.  1.   What  is  the  amount  of  20  dollars,  at  6  percent, 
compound  interest,  for  100  years  ? 

Amount  of  1  dollar  for  1  year    1.06     log.     0.0253059 
Multiplying  by  100 

2.53059 

Given  principal  20  1.30103 

Amount  required  $6786  3.83162 

2.  What  is  the  amount  of  1  cent  at  6  per  cent,  compound 
interest,  in  500  years  ? 

Amount  of  1  dollar  for  1  year    1.06     log.     0.0253059 
Multiplying  by  500 

12.65295 
Given  principal  0.01  -2.00000 

Amount  $44,973,000,000  10.65295 

More  exact  answers  may  be  obtained,  by  using  logarithms 
of  a  greater  number  of  decimal  places. 

3.  What  is  the  amount  of  1000  dollars,  at  6  per  cent, 
compound  interest,  for  10  years?  Ans,  1790.80. 


40  INCREASE    OF    POPULATION. 

4.  What  principal,  at  4  per  cent,  interest,  will  amount  to 
1643  dollars  in  21  years?  Ans.  721. 

5.  What  principal,  at  6  per  cent.,  will  amount  to  202  dol- 
lars in  4  years  ?  Ans.  160. 

6.  At  what  rate  of  interest,  will  400  dollars  amount  to 
669i,  in  9  years  ?  Ans.  4  per  cent. 

7.  In  how  many  years  will  500  dollars  amount  to  900,  at 
5  per  cent,  compound  interest?  Ans.  12  years. 

8.  In  what  time  will  10,000  dollars  amount  to  16,288,  at 
5  per  cent  compound  interest  ?  Ans.  10  years. 

9.  At  what  rate  of  interest,  will  11,106  dollars  amount  to 
20.000  in  15  years  ?  Ans.  4  per  cent. 

10.  What  principal,  at  6  per  cent,  compound  interest,  will 
amount  to  3188  dollars  in  8  years?  Ans.  $2000. 

11.  What  will  be  the  amount  of  1200  dollars,  at  6  per 
cent  compound  interest,  in  10  years,  if  the  interest  is  con- 
verted into  principal  every  half  year?        Ans.  2167.3  dolls. 

12.  In  what  time  will  a  sum  of  money  double,  at  6  per 
cent  compound  interest  ?  Ans.  11.9  years. 

13.  What  is  the  amount  of  5000  dollars,  at  6  per  cent, 
compound  interest,  for  28^-  years  ?          Ans.  25.942  dollars. 


INCREASE    OF   POPULATION. 

62.  The  natural  increase  of  population  in  a  country,  may 
be  calculated  in  the  same  manner  as  compound  interest ;  on 
the  supposition,  that  the  yearly  rate  of  increase  is  regularly 
proportioned  to  the  actual  number  of  inhabitants.  From 
the  population  at  the  beginning  of  the  year,  the  rate  of  in- 
crease being  given,  may  be  computed  the  whole  increase 
during  the  year.  This,  added  to  the  number  at  the  begin- 
ning, will  give  the  amount,  on  which  the  increase  of  the 
second  year  is  to  be  calculated,  in  the  same  manner  as  the 


INCREASE    OF   POPULATION.  41 

first  year's  interest  on  a  sum  of  money,  added  to  the  sum 
itself,  gives  the  amount  on  which  the  interest  for  the  second 
year  is  to  be  calculated. 

If  P=the  population  at  the  beginning  of  the  year, 

a=l+the  fraction  which  expresses  the  rate  of  increase, 
«=any  number  of  years ;  and 
A=the  amount  of  the  population  at  the  end  of  n  ye 
then,  as  in  the  preceding  article, 

A=a*xP,  and 

1.  Log.  A=rcXlog.  a+log.  P. 

2.  Log.  P=log.  A — wXlog.  a. 

log.  A— log.  P. 


3.  Log.  a— 

4.  n= 


n 
log.  A — log.  P. 


log.  a. 

Ex.  1.  The  population  of  the  United  States  in  1840  was 
(in  round  numbers)  1Y,OYO,000.*  Supposing  the  yearly  rate 

*  For  some  very  interesting  views  of  the  progress  of  population,  &c., 
in  the  United  States,  see  Prof.  George  Tucker's  elaborate  essays,  first 
published  in  the  Merchant's  Magazine,  1842 — 3,  and  subsequently  in  a 
separate  volume.  • 

The  following  tables  show  the  official  census  of  the  United  States 
from  1790  to  1840  with  the  decennial  rate  of  increase. 


POPULATION. 


1790.      |     1800.      |     1810.      |     1820.     |      1830.       |      1840. 
3,929,827  |  5,305,925  |  7,239,814  |  9,638,131  |  12,866,020  |  17,069,453 


DECENNIAL    INCREASE. 

1800. 

|     1810. 

1820. 

1830. 

|     1840. 

35.02 

|     36.45 

|     33.35 

|     33.26 

|     33.67 

42  INCREASE    OF  POPULATION. 

of  increase  to  be  -&  part  of  the  whole,  what  will  be  the 
population  in  1850  ? 

Here  P==l 7,070,000.     n=lO.     a=I-{~3\^*. 

And  log.  A=10Xlog.  ff+log.  (17,070,000,) 
Therefore,  A=22,810,000,  the  population  in  1850. 

2.  If  the  number  of  inhabitants  in  a  country  be  five  mil- 
lions at  the  beginning  of  a  century ;  and  if  the  yearly  rate 
of  increase  be  -§V  J  what  will  be  the  number  at  the  end  of  50 
years  ?  and  what  at  the  end  of  the  century  ? 

Ans.  25,763,000,  and  132,750,000. 

3.  If  the  population  of  a  country,  at  the  end  of  a  century, 
is  found  to  be  45,860,000 ;  and  if  the  yearly  rate  of  increase 
has  been  -riir ;  what  was  the  population  at  the  commence- 
ment of  the  century  ?  Ans.  20  millions. 

4.  The  population   of   the   United   States  in   1810  was 
7,240,000  ;  in  1820,  9,625,000.     What  was  the  annual  rate 
of  increase  between  these  two  periods,   supposing  the  in- 
crease each  year  to  be  proportioned  to  the  population  at  the 
beginning  of  the  year  ? 

log.  9,625,000— log.  7,240,000 
Here  log.  o= -— 

Therefore,  o=1.029  ;  and  TlHhr>  or  2.9  per  cent,  is  the 
rate  of  increase. 

5.  The  population  of  the  United  States  on  the  1st  August, 
1820,  was  9,638,000 — in  1830,  the  time  of  taking  the  census 
was  changed  to  the  1st  June,  and  at  that  time  the  popula- 
tion was  12,866,000. — What  was  the  annual  rate  of  increase  ? 
And  what  would  have  been  the  amount  of  population  to  be 
added  for  the  subsequent  two  months  ? 

6.  In  how  many  years,  will  the  population  of  a  country 
advance  from  two  millions  to  five  millions ;  supposing  the 
yearly  rate  of  increase  to  be  -3^  ?  Ans.  47-J-  years. 


INCREASE   OF  POPULATION.  43 

7.  If  the  population  of  a  country,  at  a  given  time,  be 
seven  millions  ;  and  if  the  yearly  rate  of  increase  be  ^th  ; 
what  will  be  the  population  at  the  end  of  35  years  ? 

8.  The   population  of  the  United   States  in  1800  was 
5,306,000.    What  was  it  in  1780,  supposing  the  yearly  rate 
of  increase  to  be  -^  ? 

9.  In  what  time  will  the  population  of  a  country  advance, 
from  four  millions  to  seven  millions,  if  the  ratio  of  increase 


10.  What  must  be  the  rate  of  increase,  that  the  population 
of  a  place  may  change  from  nine  thousand  to  fifteen  thou- 
sand, in  12  years? 

If  the  population  of  a  country  is  not  affected  by  immi- 
gration or  emigration,  the  rate  of  increase  will  be  equal  to 
the  difference  between  the  ratio  of  the  birtfis,  and  the  ratio 
of  the  deaths,  when  compared  with  the  whole  population. 

Ex.  11.  If  the  population  of  a  country,  at  any  given  time, 
be  ten  millions  ;  and  the  ratio  of  the  annual  number  of  births 
to  the  whole  population  be  -jV,  and  the  ratio  of  deaths  -jV» 
what  will  be  the  number  of  inhabitants,  at  the  end  of  60 
years? 

Here  the  yearly  rate  of  increase==^l4  —  3V=dhr. 

And  the  population,  at  the  end  of  60  years=3  1,750,000. 

The  rate  of  increase  or  decrease  from  immigration  or  emi- 
gration, will  be  equal  to  the  difference  between  the  ratio  of 
immigration  and  the  ratio  of  emigration  ;  and  if  this  differ- 
be  added  to,  or  subtracted  from,  the  difference  between  the 
ratio  of  the  births  and  that  of  the  deaths,  the  whole  rate 
of  increase  will  be  obtained. 

Ex.  12.  If  in  a  country,  the  ratio  of  births  be       -gV> 
the  ratio  of  deaths  ^, 

the  ratio  of  immigration  -5^, 
the  ratio  of  emigration     eV, 


44  INCREASE  OF  POPULATION". 

and  if  the  population  this  year  be  10  millions,  what  will  it 
be  20  years  hence  ? 

The  rate  of  the  natural  increase  =-§L0 — fa — 120  J 
That  of  increase  from  immigration  =5*0 — Jo — sio ; 
The  sum  of  the  two  is  ^fo ; 

-And  the  population  at  the  end  of  20  years,  is  12,611,000. 

13.  If  the  ratio  of  the  births  be     *V, 

of  the  deaths         -gV, 

of  immigration       -jV, 

of  emigration         -§V, 

in  what  time  will  three  millions  increase  to  four  and  a  half 
millions  ? 

If  the  period  in  which  the  population  will  double  be  given ; 
the  numbers  for  several  successive  periods,  will  evidently  be 
in  a  geometrical  progression,  of  which  the  ratio  is  2  ;  and  as 
the  number  of  periods  will  be  one  less  than  the  number  of 
terms ; 

If  P=the  first  term, 

A=the  last  term, 

?i=the  number  of  periods  ; 
Then  will  A=Px2n,  (Alg.  439.) 
Or  log.  A=log.  P+wXlog.  2. 

Ex  1 .  If  the  descendants  of  a  single  pair  double  once  in 
25  years,  what  will  be  their  number  at  the  end  of  one  thou- 
sand years  ? 

The  number  of  periods  here  is  40. 
And  A=2X240=2,199,200,000,000. 

2.  If  the  descendants  of  Noah,  beginning  with  his  three 
sons  and  their  wives,  doubled  once  in  20  years  for  300  years, 
what  was  their  number,  at  the  end  of  this  time  ? 

Ans.  196,608. 

8.  The  population  of  the  United  States  in  1820  being 


EXPONENTIAL   EQUATIONS.  45 

9,638,000 ;  what  must  it  be  in  the  year  2020,  supposing  it 
to  double  once  in  25  years  ?  Ans.  2,467,333,000. 

4.  Supposing  the  descendants  of  the  first  human  pair  to 
double  once  in  50  years,  for  1650  years,  to  the  time  of  the 
deluge,  what  was  the  population  of  the  world,  at  that  time  ? 


EXPONENTIAL   EQUATIONS. 

62.  An  EXPONENTIAL  equation  is  one  in  which  the  letter 
expressing  the  unknown  quantity  is  an  exponent. 

Thus  ax=b,  and  xx=bc,  are  exponential  equations.  These 
are  most  easily  solved  by  logarithms.  As  the  two  members 
of  an  equation  are  equal,  their  logarithms  must  also  be  equal. 
If  the  logarithm  of  each  side  be  taken,  the  equation  may 
then  be  reduced,  by  the  rules  given  in  algebra. 

Ex.  What  is  the  value  of  x  in  the  equation  3*=243  ? 

Taking  the  logarithms  of  both  sides,  log.  3ar==log.  243. 
But  the  logarithm  of  a  power  is  equal  to  the  logarithm  of 
the  root,  multiplied  into  the  index  of  the  power.  (Art.  45.) 

Therefore  (log.  3)x#=log.  243  ;  and  dividing  by  log.  3. 

log.  243     2.38561 

x=—  -  =  -  =5.     So  that  36=243. 
log  3.       0.47712 

64.  The  exponent  of  a  power  may  be  itself  a  power,  as  in 
the  equation 


where  x  is  the  exponent  of  the  power  m*,  which  is  the  ex- 

X 

ponent  of  the  power  am. 

X 

Ex.  4.  Find  the  value  of  x,  in  the  equation  98=1000. 

log.  1000. 
3*X(log.  9)—  log,  1000.  Therefore,  3*-^   Io     & 


46  EXPONENTIAL   EQUATIONS. 

Then,  as  3^=3.14.     a-(log.  3)=log.  3.14 

log.  3.14 

Therefore,  x= =±^f£fff=i.o4. 

log.  3 

In  cases  like  this,  where  the  factors,  divisors,  &c.  are  loga- 
rithms, the  calculation  may  be  facilitated,  by  taking  the 
logarithms  of  the  logarithms.  Thus  the  value  of  the  fraction 
*?7?in  is  most  easily  found,  by  subtracting  the  logarithm 
of  the  logarithm  which  constitutes  the  denominator,  from 
the  logarithm  of  that  which  forms  the  numerator. 

bax+d 
5.  Find  the  value  of  xt  in  the  equation ==m 

log.  (cm — d) — log.  6. 


log.  o. 


TRIGONOMETRY, 

SECTION  I. 

SINES,    TANGENTS,    SECANTS,    &C. 

AKT.  Vl.  TRIGONOMETRY  treats  of  the  relations  of  the 
sides  and  angles  of  TRIANGLES.  Its  first  object  is  to  deter- 
mine the  length  of  the  sides,  and  the  quantity  of  the  angles. 
In  addition  to  this,  from  its  principles  are  derived  many  in- 
teresting methods  of  investigation  in  the  higher  branches  of 
analysis,  particularly  in  physical  astronomy. 

72.  Trigonometry  is  either  plane  or  spherical.     The  for- 
mer treats  of  triangles  bounded  by  right  lines  ;  the  latter, 
of  triangles  bounded  by  arcs  of  circles. 

Divisions  of  the  Circle. 

73.  In  a  triangle  there  are  two  classes  of  quantities  which 
are  the  subjects  of  inquiry,  the  sides  and  the  angles.     For 
the  purpose  of  measuring  the  latter,  a  circle  is  introduced. 

The  periphery  of  every  circle,  whether  great  or  small,  is 
supposed  to  be  divided  into  360  equal  parts  called  degrees, 
each  degree  into  60  minutes,  each  minute  into  60  seconds, 
each  second  into  60  thirds,  &c.,  marked  with  the  characters 
°,  ',  ",  "',  &c.  Thus,  32°  24'  13"  22'"  is  32  degrees,  24  min- 
utes, 13  seconds,  22  thirds. 

A  degree,  then,  is  not  a  magnitude  of  a  given  length;  but 


48 


TRIGONOMETRY. 


a  certain  portion  of  the  whole  circumference  of  any  circle. 
It  is  evident  that  the  360th  part  of  a  large  circle  is  greater 
than  the  same  part  of  a  small  one.  On  the  other  hand,  the 
number  of  degrees  in  a  small  circle,  is  the  same  as  in  a  large 
one. 

The  fourth  part  of  a  circle  is  called  a  quadrant,  and  con- 
tains 90  degrees. 

74.  To  measure  an  angle,  a  circle  is  so  described  that  its 
center  shall  be  the  angular  point,  and  its  periphery  shall  cut 
the  two  lines  which  include  the  angle.    The  arc  between  the 
two  lines  is  considered   a  measure  of 

the  angle,  because,  by  Euc.  33.  6, 
angles  at  the  center  of  a  given  circle, 
have  the  same  ratio  to  each  other,  as  the 
arcs  on  which  they  stand.  Thus  the 
arc  AB,  is  a  measure  of  the  angle 
ACB. 

It  is  immaterial  what  is 
the  size  of  the  circle,  pro- 
vided it  cuts  the  lines 
which  include  the  angle. 
Thus,  the  angle  ACD  is 
measured  by  either  of 
the  arcs  AG,  ag.  For 
ACD  is  to  ACH,  as  AG 
to  AH,  or  as  ag  to  ah.  (Euc.  33.  6.) 

75.  In  the   circle  ADGH,  let    the 
two  diameters  AG  and  DH  be  perpen- 
dicular  to  each    other.      The   angles 
ACD,  DCG,    GCH,   and  HCA,  will 
be  right  angles;    and  the   periphery 
of  the  circle  will  be  divided  into  four 
equal     parts,     each     containing     90 

degrees.  As  a  right  angle  is  subtended  by  an  arc  of 
90°,  the  angle  itself  is  said  to  contain  90°.  Hence,  in  two 


SINES,    TANGENTS,    <feC.  49 

right  angles,  there  are  180°;  in  four  right  angles,  360°; 
and  in  any  other  angle,  as  many  degrees,  as  in  the  arc  by 
which  it  is  subtended. 

76.  The  sum  of  the  three  angles  of  any  triangle  being 
equal  to  two  right  angles,  (Euc.  32.  1.*)  is  equal  to  180°. 
Hence,  there  can  never  be  more  than  one  obtuse  angle  in  a 
triangle.     For  the  sum  of  two  obtuse  angles  is  more  than 
180°. 

77.  The  COMPLEMENT  of  an  arc  or  an  a-rigle,  is  the  differ- 
mce  between  the  arc  or  angle  and  90  degrees. 

The  complement  of  the  arc  AB  is  DB  ;  and  title  comple- 
ment of  the  angle  ACB  is  DOB.  The  complement  of  the 
»rc  BDG  is  also  DB, 

The  complement  of  10°  is  80°,  of  60°  is  30°, 
of  20°  is  70°,  of  120°  is  30°, 
of  50°  is  40°,  of  170°  is  80°,  &c, 


Hence,  an  acute  angle  and  its  com- 
plement are  always  equal  to  90°. 
The  angles  ACB  and  DCB  are  to- 
gether equal  to  a  right  angle.  The 
two  acute  angles  of  a  right  angled 
triangle  are  equal  to  90° :  therefore 
each  is  the  complement  of  the  other. 

78.   The  SUPPLEMENT  of  an  arc  or  an  angle  is  the  differ- 
ence between  the  arc  or  angle  and  180  degrees. 

The  supplement  of  the  arc  BDG  is  AB ;  and  the  supple- 
ment of  the  angle  BCG  is  BOA. 

The  supplement  of  10°  is  170°,       of  120°  is  60°, 

of  80°  is  100°,       of  150°  is  30°,  &c. 

Hence  an  angle  and  its  supplement  are  always  equal  to 

*  Thomson's  Geometry,  38,  1. 
5 


50  TRIGONOMETRY.       . 

180o.     The  angles  BOA  and  BOG  are  together  equal  to  two 
right  angles. 

79.  Cor.  As  the  three  angles  of  a  plane  triangle  are  equal 
to  two  right  angles,  that  is,  to  180°  (Euc.  32.  1.)  the  sum 
of  any  two  of  them  is  the  supplement  of  the  other.     So 
that  the  third  angle  may  be  found,  by  subtracting  the  sum 
of  the  other  two  from  1CO°.     Or  the  sum  of  any  two  may 
be  found,  by  subtracting  the  third  from  180°. 

80.  A  straight  line  drawn  from  the  centre  of  a  circle  to 
any  part  of  the  periphery,  is  called  a  radius  of  the  circle, 
In  many  calculations,  it  is  convenient  to  consider  the  radius, 
whatever  be  its  length,  as  a  unit.  (Alg.  510.)     To  this  must 
be  referred  the  numbers  expressing  the  lengths  of  other 
lines.     Thus,  20  will  be  twenty  times  the  radius,  and  0.75, 
three-fourths  of  the  radius. 


Definitions  of  Sines,  Tangents,  Secants,  <&c. 

81.  To  facilitate  the  calculations  in  Trigonometry,  there 
are  drawn,  within  and  about  the  circle,  a  number  of  straight 
lines,  called  Sines,  Tangents,  Secants,  <6c.     With  these  the 
learner  should  make  himself  perfectly  familiar.     The  direct 
and  proper  measure  of  an  angle  is  an  arc  of  a  circle.  (Art.  74.) 
But  trigonometrical  solutions  are  commonly  made  with  the 
aid  of  certain  straight  lines,  which  have  known  relations  to 
the  arcs  to  which  they  belong. 

82.  The  SINE  of  an  arc  is  a  straight  line  drawn  from 
one  end  of  the  arc,  perpendicular  to  a  diameter  which  passes 
through  the  other  end. 

Thus,  BG  is  the  sine  of  the  arc  AG.  For  BG  is  a  line 
drawn  from  the  end  G  of  the  arc,  perpendicular  to  the  diam- 
eter AM  which  passes  through  the  other  end  A  of  the  arc. 

Cor.  The  sine  is  half  the  chord  of  double  the  arc.  The 
sine  BG  is  half  PG,  which  ia  the  chord  of  the  arc  PAG, 
double  the  arc  AG. 


SINES,    TANGENTS,   AC. 


51 


83.  The  VEBSED  SINE 
of  an  arc  is  that  part  of 
the   diameter    which     is 
between  the  sine  and  the 
arc. 

Thus,  BA  is  the  versed 
sine  of  the  arc  AG. 

84.  The  TANGENT  of 
an  arc,  is  a  straight  line 
drawn     perpendicularly 
from  the  extremity  of  the 
diameter    which     passes 

through  one  end  of  the  arc,  and  extended  till  it  meets  a  line 
drawn  from  the  centre  through  the  other  end. 
Thus,  AD  is  the  tangent  of  the  arc  AG. 

85.  The  SECANT  of  an  arc  is  a  straight  line  draim  from 
the  centre,  through  one  end  of  the  arc,  and  extended  to  the 
tangent  which  is  drawn  from  the  other  end. 

Thus  CD  is  the  secant  of  the  arc  AG. 

86.  In  Trigonometry,  the  terms  tangent  and  secant  have 
a  more  limited  meaning,  than  in  Geometry.    In  both,  indeed, 
the  tangent  touches  the  circle,  and  the  secant  cuts  it.     But 
in    Geometry,  these  lines   are   of  no   determinate    length; 
whereas,  in  Trigonometry,  they  extend  from  the  diameter  to 
the  point  in  which  they  intersect  each  other. 

87.  The  lines  just  defined  are  sines,  tangents,  and  se- 
cants of  arcs.     BG    is  the  sine  of  the  arc  AG.     But  this 
arc  subtends  the^ngle  GCA.     BG  is  then  the  sine  of  the 
arc  which  subtends  the  angle   GCA.     This   is    more  con- 
cisely expressed,  by  saying  that  BG  is  the  sine  of  the  angle 
GCA.     And  universally,  the  sine,  tangent,  and  secant  of  an 
arc,  are  said  to  be  the  sine,  tangent,  and  secant  of  the  angle 
which  stands  at  the  centre  of  the  circle,  and  is  subtended  by 
the  arc.     Whenever,   therefore,  the  sine,  tangent,  or  secant 
of  an  angle  is  spoken  of ;  we  are  to  suppose  a  circle  to  be 


52  TRIGOMOMETRY. 

drawn  whose  centre  is  the  angular  point ;  and  that  the  lines 
mentioned  belong  to  that  arc  of  the  periphery  which  sub- 
tends the  angle. 

88.  The  sine  and  tangent  of  an  acute  angle,  are  opposite 
to  the  angle.     But  the  secant  is  one  of  the  lines  which  in- 
clude the  angle.     Thus,  the  sine  BG,  and  the  tangent  AD, 
are  opposite  to  the  angle  DC  A.     But  the  secant  CD  is  one 
of  the  lines  which  include  the  angle. 

89.  The  sine  complement  or  COSINE  of  an  angle,  is  the  sine 
of  the  COMPLEMENT  of  that  angle.     Thus,   if  the  diameter 
HO  be  perpendicular  to  MA,  the  angle  HCG  is  the  com- 
plement of  ACG ;    (Art.   77.)  and  LG,   or  its  equal  CB, 
is  the  sine  of  HCG.  (Art.  82.)   It  is,  therefore,  the  cosine  of 
GCA.     On  the  other  hand,  GB  is  the  sine  of  GCA,  and  the 
cosine  of  GCH. 

So  also  the  cotangent  of  an  angle  is  the  tangent  of  the 
complement  of  the  angle.  Thus,  HF  is  the  cotangent  of 
GCA.  And  the  cosecant  of  an  angle  is  the  secant  of  the 
complement  of  the  angle.  Thus,  CF  is  the  cosecant  of  GCA. 

Hence,  as  in  a  right  angled  triangle,  one  of  the  acute 
angles  is  the  complement  of  the  other;  (Art.  77.)  the  sine, 
tangent,  and  secant  of  one  of  these  angles,  are  the  cosine, 
cotangent,  and  cosecant  of  the  other. 

90.  The  sine,  tangent,  and  secant  of  the  supplement  of  an 
angle,  are  each  equal  to  the  sine,  tangent,  and  secant  of  the 
angle  itself.     It  will  be  seen,  by  applying  the  definition.  (Art 
82.)  to  the  figure,,  that  the  sine  of  the  obtuse  angle  GCM  is 
BG,  which  is  also  the  sine  of  the  acute  angle  GCA.     It 
should  be  observed,  however,  tiat  the  sine  of  an  acute  angle 
is  opposite  to  it ;  while  the   sine  of  an  obtuse  angle  falls 
without  the  angle,  and  is  opposite  to  its  supplement.     Thus 
BG,  the  sine  of  the  angle  MCG,  is  not  opposite  to  MCG, 
but  to  its  supplement  ACG. 

The  tangent  of  the  obtuse  angle  MCG  is  MT,  or  its  equal 


SINES,    TANGENTS,    <feC.  53 

AD,  which  is  also  the  tangent  of  ACG.     And  the  secant  of 
MCG  is  CD,  which  is  also  the  secant  of  ACG. 

91.  But  the  versed+sine  of  an  angle  is  not  the  same  as  that 
of  its  supplement.     The  versed  sine  of  an  acute  angle  is 
equal  to  the  difference  between  the  cosine  and  radius.     But 
the  versed  sine  of  an  obtuse  angle  is  equal  to  the  sum  of 
the  cosine  and  radius.     Thus,  the  versed  sine  of  ACG  is 
AB=AC— BC.  (Art.  83.)     But  the  versed  sine  of  MCG  is 
MB=MC+BC. 

Relations  of  Sines,  Tangents,  Secants,  &c.,  to  each  other. 

92.  The  relations  of  the  sine,  tangent,  secant,  cosine,  &c., 
to  each  other,  are  easily  derived  from  the  proportions  of  the 
sides  of  similar  triangles.   (Euc.  4.  6.*)      In  the  quadrant 
ACH,  these   lines  form 

three  similar  triangles, 
viz.  ACD,BCGorLCG, 
and  HCF.  For,  in  each 
of  these,  there  is  one 
right  angle,  because  the 
sines  and  tangents  are, 
by  definition,  perpen- 
dicular to  AC  ;  as  the 
cosine  and  cotangent  are 
to  CH.  The  lines  CH, 
BG,  and  AD,  are  paral- 
lel, because  C A  makes  a  right  angle  with  each.  (Euc.  27.  l.f) 
For  the  same  reason,  CA,  LG,  and  HF,  are  parallel.  The 
alternate  angles  GCL,  BGC,  and  the  opposite  angle  CDA, 
are  equal;  (Euc.  29.  l.f)  as  are  also  the  angles  GCB,  LGC, 
and  HFC.  The  triangles  ACD,  BCG,  and  &CF,  are  there- 
fore similar. 


*  Thomson,  18.  4. 


t  Ibid.  18.  1. 
5* 


J  Ibid.  24. 1. 


TRIGONOMETRY. 


It  should  also  be  observed,  that  the  line  BC,  between  the 
sine  and  the  centre  of  the  circle,  is  parallel  and  equal  to  the 
cosine ;  and  that  LC,  between  the  cosine  and  centre,  is  par- 
allel and  equal  to  the  sine  ;  (Euc.  34. 1.*)  so  that  one  may  be 
taken  for  the  other  in  any  calculation. 

93.  From  these  sim- 
ilar triangles,  are  de- 
rived the  following  pro- 
portions ;  in  which  R  is 
put  for  radius, 

sin  for  sine, 
cos  for  cosine, 
tan  for  tangent, 
cot  for  cotangent, 
sec  for  secant, 
cosec  for  cosecant. 

By  comparing  the  triangles  CBG  and  CAD, 

1.  AC      BC  :  :  AD      BG,  that  is,  R  :  cos  :  :  tan  :  sin. 

2.  CG      CD  :  :  BG      AD  R  :  sec  :  :  sin  :  tan. 
8.  CB     CA  :  :  CG      CD               cos  :  R  :  :  R  :  see. 

Therefore  R*=cosXsec. 

By  comparing  the  triangles  CLG  and  CHF, 

4.  Cfl     CL  :  :  HF     LG,  that  is,  R  :  sin  ::  cot :  cos. 

5.  CG      CF  :  :  LG      HF,  R  :  cosec  :  :  cos  :  cot. 

6.  CL      CH  :  :  CG     CF  sin  :  R  :  :  R  :  cosec. 

Therefore  Ra=sin  X  cosec. 
By  comparing  the  triangles  CAD  and  CHF, 


7.  CH 

AD 

::CF 

CD,  that  is,  R  : 

tan  :  :  cosec  :  sec. 

8.  CA 
9.  AD 

HF 
Afc 

:  :  CD 
::CH 

CF              R  : 
HF              tan 

cot  :  :  sec  ;  cosec. 
I  R  :  :  R  :  cot. 

Therefore  R8= 

=tanXcot. 

*  Thomson,  26.  1. 


•IFB8,    TANGENTS,    AC.  ££ 

It  will  not  be  necessary  for  the  learner  to  commit  these 
proportions  to  memory.  But  he  ought  to  make  himself  so 
familiar  with  the  manner  of  stating  them  from  the  figure,  as 
to  be  able  to  explain  them,  whenever  they  are  referred  to. 

94.  Other  relations  of  the  sine,  tangent,  &c.,  may  be  de- 
rived from  the  proposition,  that  the  square  of  the  hypothe- 
nuse  is  equal  to  the  sum  of  the  squares  of  the  perpendicular 
sides.  (Euc.  47.  1.—  Thomson  11.  4.) 

In  the  right  angled  triangles  CBG,  CAD,  and  CHF, 

1.  CG2=CB"a-f-BG:a,  that  is,  R2=cos2+sin2,* 

2.  CD3=CA9-f  AD8  sec*=Ra+tan1, 

3.  CFa=CHa4-HFa  coseca=Ra+cot3, 


And,  extracting  the  root  of  both  sides,  (Alg.  296.) 


E>=V  cos*+sin*= V  sec2 — tan2=V  cosec2 — cot2 
Hence,  if  R=l,  (Alg  385.) 


Sin=V  1— cos2  Sec=V  1  +tana 

Cos=V  1 — sin1  Cosec=Vl-fcot* 

95.  The  sine  of  90C 


The  chord  of  GO 


are,  in  any  circle,  each  equal 


And  the  tangent  of  45° 
to  the  radius,  and  therefore  equal  to  each  other . 

Demons  tration. 

1.  In  the  quadrant  ACH,  (figure  on  the  next  page,)  the 
arc  AH  is  90°.  The  sine  of  this,  according  to  the  definition, 
(Art.  82.)  is  CH,  the  radius  of  the  circle. 


*  Sin2  is  here  put  for  the  square  of  the  sine,  cos2  for  the  square  of  the 
cosine,  &c. 


56  TRIGONOMETRY. 

2.  Let  AS  be   an  arc  of  60°. 
Then  the  angle  ACS,  being  mea- 
sured by  this  arc,  will  also  con- 
tain 60° ;  (Art.  75.)  and  the  tri- 
angle  ACS  will    be   equilateral. 
For  the  sum  of  the  three  angles 
is     equal    to    180°.     (Art.    76.) 
From  this,  taking  the  angle  ACS, 
which  is  60°,  the  sum  of  the  re- 
maining two  is  120°.     But  these  two  are  equal,  because  they 
are  subtended  by  the  equal  sides,  CA  and  CS,  both  radii  of 
the  circle.     Each,  therefore,  is  equal  to  half  120°,  that  is,  to 
60°.  All  the  angles  being  equal,  the  sides  are  equal,  and  there- 
fore AS,  the  chord  of  60°,  is  equal  to  CS,  the  radius. 

3.  Let  AR  be  an  arc  of  45°.     AD  will  be  its  tangent,  and 
the  angle  ACD  subtended  by  the  arc,  will  contain  45°.    The 
angle  CAD  is  a  right  angle,  because  the  tangent  is,  by  defi- 
nition, perpendicular  to  the  radius  AC.    (Art.   84.)     Sub- 
tracting ACD,  which  is  45°,  from  90°,  (Art.  77.)  the  other 
acute  angle  ADC  will  be  45°  also.     Therefore  the  two  legs 
of  the  triangle  ACD  are  equal,  because  they  are  subtended 
by  equal  angles  ;  (Euc.  6.  1.)  that  is,  AD  the  tangent  of  45°, 
is  equal  to  AC  the  radius. 

Cor.  The  cotangent  of  45°  is  also  equal  to  radius.  For 
the  complement  of  45°  is  itself  45°.  Thus,  HD,  the  cotan- 
gent of  ACD,  is  equal  to  AC  the  radius. 

96.  The  sine  of  30°  is  equal  to  half  radius.  For  the 
sine  of  30°  is  equal  to  half  the  chord  of  60°.  (Art.  82.  cor.) 
But  by  the  preceding  article,  the  chord  of  60°  is  equal  to 
radius.  Its  half,  therefore,  which  is  the  sine  of  30°,  is  equal 
to  half  radius. 

Cor.  1.  The  cosine  of  60°  is  equal  to  half  radius.  For 
the  cosine  of  60°  is  the  sine  of  30°.  (Art.  89.) 

Cor.  2.     The  cosine  of  30°=£V3.     For 

Cos2  30°=R2— sin2  30°=1 — i=f. 


SINES,  TANGENTS,  AO.  57 

Therefore, 

Cos  30°=vi=i-V3. 

1 

96.  b.  The  sine  of  45°== — •  For 

V2 

RJ=l=sina  45°+cosa  45=2  sin9  45° 

1 

Therefore,  Sin  45°=Vi= — • 
V2 

97.  The  chord  of  any  arc  is  a  mean  proportional,  between 
the  diameter  of  the  circle,  and  the  versed  sine  of  the  arc. 

Let  ADB,  be  an  arc,  of  which 
AB  is  the  chord,  BF  the  sine, 
and  AF  the  versed  sine.  The 

Tl 

angle  ABH  is  a  right  angle,  (Euc. 
31.  3.*)  and  the  triangles  ABH, 
and  ABF,  are  similar.  (Euc.  8.  6.f) 
Therefore, 

AH  :  AB  :  :  AB  :  AF. 

That  is,  the  diameter  is  to  the  chord,  as  the  chord  to  the 
versed  sine. 

Let  the  arc  AD=a,  and  ADB=2a.  Draw  BF  perpen- 
dicular to  AH.  This  will  divide  the  right  angled  triangle 
ABH  into  two  similar  triangles.  (Euc.  8.  6.)  The  angles 
ACD  and  AHB  are  equal.  (Euc.  20.  3.J)  Therefore  the 
four  triangles  ACG,  AHB,  FHB,  and  FAB  are  similar ;  and 
the  line  BH  is  twice  CG,  because  BH  :  CG  :  :  HA  :  CA. 

The  sides  of  the  four  triangles  are, 

AG=sin  a,         CG=cos  a.         HF=vers.  sup.  2o, 
AB=2  sin  a,  •    BH=2  cos  a.     AC=the  radius, 
BF=sin  2a,        AF=vers  2a,     AH=the  diameter. 

*  Thomson,  13.  2.    Cor.  2.  f  Ibid.  22.  4.  J:  Ibid.  13.  9. 


ONOMETRY. 

A  variety  of   proportions  may  be  stated,  between  the 
homologous  sides  of  these  triangles  :  For  instance, 

By  comparing  the  triangles  ACG  and  ABF, 
AC  :  AG  :  :  AB  :  AF,  that  is,  R  :  sin  a  : :  2  sin  a  ;  vers  2a 
AC  :  CG  :  :  AB  :  BF,  R  :  cosa : :  2  sin  a  :  sin  2a 

AG  :  CG  :  :  AF  :  BF,  Sin  a  :  cos  a  :  :  vers  2a  :  sin  2a 


Therefore, 


2a=2sinaa 
2o=2sin  aXcos  a 
2a=vers  2aXcos  a 


Sin 


By    comparing    the    triangles 
ACG  and  BFH, 


AC  ;  CG  :  :  BH  :  HF,  that  is,  R  :  cos  a  :  :  2cos  a  :  vers.  sup.  2a 
AG  :  CG  :  :  BF  :  HF,       Sin  a  :  cos  a  :  :  sin  2a  :  vers.  sup.  2a 

Therefore, 

BXvers.  sup.  2a=2  cosaa 
Sin  a  X  vers.  sup.  2a=cos  aXsin  2a 
&c.  <fec. 

That  is,  the  product  of  radius  into  the  versed  sine  of  the 
supplement  of  twice  a  given  arc,  is  equal  to  twice  the  square 
of  the  cosine  of  the  arc. 

And  the  product  of  the  sine  of  an  arc,  into  the  versed 
sine  of  the  supplement  of  twice  the  arc,  is  equal  to  the  pro- 
duct of  the  cosine  of  the  arc,  into  the  sine  of  twice  the 
arc,  <fec.,  <fec. 


THE    TRIGONOMETRICAL    TABLES. 


SECTION  II. 

THE   TRIGONOMETRICAL   TABLES. 

ART.  98.  To  facilitate  the  operations  in  trigonometry,  the 
sine,  tangent,  secant,  &c.,  have  been  calculated  for  every 
degree  and  minute,  and  in  some  instances,  for  every  second, 
of  a  quadrant,  and  arranged  in  tables.  These  constitute 
what  is  called  the  Trigonometrical  Canon.  It  is  not  neces- 
sary to  extend  these  tables  beyond  90° ;  because  the  sines, 
tangents,  and  secants,  are  of  the  same  magnitude,  in  one  of 
the  quadrants  of  a  circle,  as  in  the  others.  Thus  the  sine  of 
30°  is  equal  to  that  of  150°.  (Art  90.) 

99.  And  in  any  instance,  if  we  have  occasion  for  the  sine, 
tangent,  or  secant  of  an  obtuse  angle,  we  may  obtain  it,  by 
looking  for  its  equal,  the  sine,  tangent,  or  secant  of  the  sup- 
plementary acute  angle^ 

100.  The  tables  are  calculated  for  a  circle  whose  radius  is 
supposed  to  be  a  unit.     It  may  be  an  inch,  a  yard,  a  mile, 
or  any  other  denomination  of  length.     But  the  sines,  tan- 
gents,  &c.,  must  always  be  understood  to  be  of  the  same  de- 
nomination as  the  radius. 

101.  All  the  sines,  except  that  of  90°,  are  less  than  radius, 
(Art.  82.)  and  are  expressed  in  the  tables  by  decimals. 

Thus  the  sine  of  20°  is  0.34202,      of  60°  is  0.86603, 

of  40°  is  0.64279,      of  89°  is  0.99985,  &c. 

When  the  tables  are  intended  to  be  very  exact,  the  decimal 
is  carried  to  a  greater  number  of  places. 

The  tangents  of  all  angles  less  than  45°  are  also  less  than 
radius.  (Art.  95.)  But  the  tangejits  of  angles  greater  than 
45°,  are  greater  than  radius,  and  are  expressed  by  a  whole 


60  THE   TRIGONOMETRICAL   TABLES. 

number  and  a  decimal.  It  is  evident  that  all  the  secants  also 
must  be  greater  than  radius,  as  they  extend  from  the  centre, 
to  a  point  without  the  circle. 

102.  The  numbers  in  the  table  here  spoken  of,  are  called 
natural  sines,  tangents,  &c.     They  express  the  lengths  of 
the  several  lines  which  have  been  defined  in  Arts.  82,  «-?, 
&c.     By  means  of  them,  the  angles  and  sides  of  triangles 
may  be  accurately  determined.     But  the  calculations  must 
be  made  by  the   tedious  processes  of   multiplication   and 
division.     To  avoid  this  inconvenience,  another  set  of  tables 
has  been  provided,  in  which  are  inserted  the  logarithms  of 
the  natural  sines,  tangents,  £c.     By  the  use  of  these,  ad- 
dition and  subtraction  are  made  to  perform  the  office  of  mul- 
tiplication and  division.     On  this  account,  the  tables  of  loga- 
rithmic, or  as  they  are  sometimes   called,  artificial   sines, 
tangents,   &c.,  are  much  more  valuable,  for  practical  pur- 
poses, than  the  natural  sines,  &c.     Still  it  must  be  remem- 
bered that  the  former  are  derived  from  the  latter.    The  arti- 
ficial sine  of  an  angle,  is  the  logarithm  of  the  natural  sine 
of  that  angle.     The  artificial  tangent  is  the  logarithm  of  the 
natural  tangent,  <fec. 

103.  One  circumstance,  -  however,  is  to  be  attended  to,  in 
comparing  the  two  sets  of  tables.     The  radius  to  which  the 
natural  sines,  &c.,  are  calculated,  is  unity.  (Art.  100.)    The 
secants,  and  a  part  of  the  tangents  are,  therefore,  greater 
than  a  unit ;  while  the  sines,  and  another  part  of  the  tan- 
gents, are  less  than  a  unit.     When  the  logarithms  of  these 
are  taken,  some  of  the  indices  will  be  positive,  and  others 
negative  ;  (Art.  9.)  and  the  throwing  of  them  together  in  the 
same  table,  if  it  does  not  lead  to  error,  will  at  least  be  at- 
tended with  inconvenience.     To  remedy  this,  10  is  added  to 
each  of  the  indices.  (Art.  12.)     They  are  then  all  positive. 
Thus  the  natural  sine  of  20°  is  0.34202.     The  logarithm  of 
this  is  "L53405.     But  the  index,  by  the  addition  of  10,  be- 


THE   TRIGONOMETRICAL   TABLES.  61 

comes    10 — 1=9.     The  logarithmic  sine  in   the  tables   is 
therefore  9.53405.* 


Directions  for  taking  Sines,  Cosines,  &c.,from  the  tables. 

104.  The  cosine,  cotangent,  and  cosecant  of  an  angle,  are 
the  sine,  tangent,  and  secant  of  the  complement  of  the  angle. 
(Art.  89.)     As  the  complement  of  an  angle  is  the  differ- 
ence between  the  angle  and  90°,  and  as  45  is  the  half  of  90  ; 
if  any  given  angle  within  the  quadrant  is  greater  than  45°, 
its  complement  is  less  ;  and,  on  the  other  hand,  if  the  angle 
is  less  than  45°,  its  complement  is  greater.     Hence,  every 
cosine,  cotangent,  and  cosecant  of  an  angle  greater  than  45°, 
has  its  equal  among  the  sines,  tangents,  and  secants  of  angles 
less  than  45°,  and  v.  v. 

Now,  to  bring  the  trigonometrical  tables  within  a  small 
compass,  the  same  column  is  made  to  answer  for  the  sines 
of  a  number  of  angles  above  45°,  and  for  the  cosines  of  an 
equal  number  below  45°. 

Thus  9.23967  is  the  log.  sine  of  10°,  and  the  cosifie  of  80°, 
9.53405      the  sine  of  20°,  and  the  cosine  of  70°,  &c. 

The  tangents  and  secants  are  arranged  in  a  similar  man- 
ner. Hence, 

105.  To  find  the  Sine,  Cosine,  Tangent,  <&c.,  of  any  num- 
ber of  degrees  and  minutes. 

If  the  given  angle  is  less  than  45°,  look  for  the  degrees 
at  the  top  of  the  table,  and  the  minutes  on  the  left ;  then, 
opposite  to  the  minutes,  and  under  the  word  sine  at  the 
head  of  the  column,  will  be  found  the  sine ;  under  the  word 
tangent,  will  be  found  the  tangent,  &c. 

*  Or  the  tables  may  be  supposed  to  be  calculated  to  the  radios 
10000000000,  whose  logarithm  is  10. 


62  THE   TRIGONOMETRICAL   TABLES. 

Thelog.  sin  of  43°  25' is  9.83715  The  tan  of  17°  20' is  9.49430 
of  17°  20'  9.47411  of  8°  46'  9.18812 

The  cos  of  17°  20'  9.97982  The  cot  of  17°  20'  10.50570 
of  8°  46'  9.99490  of  8°  46'  10.81188 

The  first  figure  is  the  index ;  and  the  other  figures  are  the 
decimal  part  of  the  logarithm. 

106.  If  the  given  angle  is  between  45°  and  90° ;  look  for 
the  degrees  at  the  bottom  of  the  table,  and  the  minutes  on 
the  right ;  then,  opposite  to  the  minutes,  and  over  the  word 
sine  at  the  foot  of  the  column,  will  be  found  the  sine ;  over 
the  word  tangent,  will  be  found  the  tangent,  &c. 

Particular  care  must  be  taken,  when  the  angle  is  less  than 
45°,  to  look  for  the  title  of  the  column,  at  the  top,  and  for 
the  minutes  on  the  left ;  but  when  the  angle  is  between  45° 
and  90°,  to  look  for  the  title  of  the  column  at  the  bottom, 
and  for  the  minutes,  on  the  right. 

The  log.  sine       of  81°  21'  is  9.99503 
The  cosine  of  72°  10'       9.48607 

The  tangent        of  54°  40'      10.14941 
The  cotangent    of  63°  22'       9.70026 

107.  If  the  given  angle  is  greater  than  90°,  look  for  the 
sine,  tangent,  &c.,  of  its  supplement.  (Art.  98,  99.) 

The  log.  sine  of  96°  44'  is  9.99699 

The  cosine  of  171°  16'       9.99494 

The  tangent  of  130°  26'      10.06952 

The  cotangent  of  156°  22'      10.35894 

108.  To  find  the  sine,  cosine,  tangent,  &c.,  of  any  number 
of  degrees,  minutes,  and  SECONDS. 

In  the  common  tables,  the  sine,  tangent,  &c.,  are  given 
only  to  every  minute  of  a  degree.*  But  they  may  be  found 
to  seconds,  by  taking  proportional  parts  of  the  difference  of 

*  In  the  very  valuable  tables  of  Michael  Taylor,  the  sines  and  tan- 
gents are  given  to  every  second. 


THE    TRIGONOMETRICAL   TABLES.  63 

the  numbers  as  they  stand  in  the  tables.  For,  -within  a  sin- 
gle minute,  the  variations  in  the  sine,  tangent,  <fcc.,  are 
nearly  proportional  to  the  variations  in  the  angle.  Hence, 

To  find  the  sine,  tangent,  &c.,  to  seconds :  Take  out  the 
number  corresponding  to  the  given  degree  and  minute ;  and 
also  that  corresponding  to  the  next  greater  minute,  and  find 
their  difference.  Then  state  this  proportion ; 

As  60,  to  the  given  number  of  seconds ; 

So  is  the  difference  found,  to  the  correction  for  the  seconds. 

This  correction,  in  the  case  of  sines,  tangents,  and  secants, 
is  to  be  added  to  the  number  answering  to  the  given  degree 
and  minute  ;  but  for  cosines,  cotangents,  and  cosecants,  the 
correction  is  to  be  subtracted  ; 

For,  as  the  sines  increase,  the  cosines  decrease. 

Ex.  1.  What  is  the  logarithmic  sine  of  14°  43'  10"  ? 

The  sine  of  14°  43'  is  9.40490 
of  14°  44'      9.40538 

Difference       48 

Here  it  is  evident  that  the  sine  of  the  required  angle  is 
greater  than  that  of  14°  43',  but  less  than  that  of  14°  44'. 
And  as  the  difference  corresponding  to  a  whole  minute  or 
60"  is  48  ;  the  difference  for  10"  must  be  a  proportional 
part  of  48.  That  is, 

60"  :  10"  :  :  48  :  8 
the  correction  to  be  added  to  the  sine  of  14°  43'. 

Therefore  the  sine  of  14°  43'  10"  is  9.40498. 

2.  What  is  the  logarithmic  cosine  of  32°  16'  45"  ? 

The  cosine  of  32°  16'  is  9.92715 
of  32°  17'  9.92707 
Difference  8 

Then,  60'' ;  45"  :  :  8  :  6  the  correction  to  be  subtracted 
from  the  cosine  of  32°  16'. 


64  THE   TRIGONOMETRICAL    TABLES. 

Therefore  the  cosine  of  32°  16'  45"  is  9.92709. 

The  tangent  of  24°  15'  18"  is     9.65376 

The  cotangent  of  31°  50'    5"  is  10.20700 

The  sine  of  58°  14'  32"  is     9.92956 

The  cosine  of  55°  10'  26"  is     9.75670 

If  the  given  number  of  seconds  be  any  even  part  of  60, 
as  i>  "3*  i>  &c.,  the  correction  may  be  found,  by  taking  a  like 
part  of  the  difference  of  the  numbers  in  the  tables,  without 
stating  a  proportion  in  form. 

109.  To  find  the  degrees  and  minutes  belonging  to  any 
given  sine,  tangent,  &c. 

This  is  reversing  the  method  of  finding  the  sine,  tangent, 
&c,.  (Art.  105,*6,  7.) 

Look  in  the  column  of  the  same  name,  for  the  sine,  tan- 
gent, &c.,  which  is  nearest  to  the  given  one  ;  and  if  the  title 
be  at  the  head  of  the  column,  take  the  degrees  at  the  top  of 
the  table,  and  the  minutes  on  the  left ;  but  if  the  title  be  at 
the  foot  of  the  column,  take  the  degrees  at  the  bottom,  and 
the  minutes  on  the  right. 

Ex.  1.  What  is  the  number  of  degrees  and  minutes  be- 
longing to  the  logarithmic  sine  9.62863? 

The  nearest  sine  in  the  tables  is  9.62865.  The  title  of 
sine  is  at  the  head  of  the  column  in  which  these  numbers  are 
found.  The  degrees  at  the  top  of  the  page  are  25,  and  the 
minutes  on  the  left  are  10.  The  angle  required  is,  therefore 
25°  10'. 

The  angle  belonging  to 

the  sine  9.87993  is  49°  20'  the  cos  9.97351  is  19°  48' 
the  tan  9.97955  43°  39'  the  cotan  9.75791  60°  12' 
the  sec  10.65396  77°  11'  the  cosec  10.49066  18°  51' 

110.  To  find  the  degrees,  minutes,  and  SECONDS,  belonging 
to  any  given  sine,  tangent,  t&c. 


THE   TRIGONOMETRICAL   TABLES.  65 

This  is  reversing  the  method  of  finding  the  sine,  tangent, 
<fcc.,  to  seconds.  (Art.  108.) 

First  find  the  difference  between  the  sine,  tangent,  &c., 
next  greater  than  the  given  one,  and  that  which  is  next  less ; 
then  the  difference  between  this  less  number  and  the  given 
one;  then 

As  the  difference  first  found,  is  to  the  other  difference ; 

So  are  60  seconds,  to  the  number  of  seconds,  which,  in  the 
case  of  sines,  tangents,  and  secants,  are  to  be  added  to  the 
degrees  and  minutes  belonging  to  the  least  of  the  two  num- 
bers taken  from  the  tables ;  but  for  cosines,  cotangents,  and 
cosecants  are  to  be  subtracted. 

Ex.  1 .  What  are  the  degrees,  minutes,  and  seconds,  be- 
longing to  the  logarithmic  sine  9.40498  ? 

Sine  next  greater  14°  44'  9.40538         Given  sine  9.40498 
Next  less       14°  43'  9.40490         Next  less    9.40490 

Difference  48         Difference  8 

Then,  48  :  8  :  :  60"  :  10",  which  added  to  14°  43',  gives 
14°  43'  10"  for  the  answer. 

2.  What  is  the  angle  belonging  to  the  cosine  9.09773  ? 

Cosine  next  greater  82°  48'  9.09807     Given  cosine  9.09773 

Next  less       82°  49'  9.09707     Next  less        9.09707 

Difference  100     Difference  66 

Then,  100  :  66  :  :  60"  :  40",  which  subtracted  from  82° 
49',  gives  82°  48'  20"  for  the  answer. 

It  must  be  observed  here,  as  in  all  other  cases,  that  of  the 
two  angles,  the  less  has  the  greater  cosine. 

The  angle  belonging  to 

the  sin  9.20621  is   9°  15'    6''  the  tan  10.43434  is  69°  48'  16" 
the  cos  9.98157     16°  34'  30"  the  cot  10.33554     24°  47'  16" 

6* 


00  THE    TRIGONOMETRICAL   TABLES. 

Method  of  Supplying  the  Secants  and  Cosecants. 

111.  In  some  trigonometrical  tables,  the  secants  and  cose- 
cants are  not  inserted.     But  they  may  be  easily  obtained 
from  the  sines  and  cosines.     For,  by  Art.  93,  proportion  3d, 

cosXsec=R2. 

That  is,  the  product  of  the  cosine  and  secant,  is  equal  to 
the  square  of  radius.  But,  in  logarithms,  addition  takes  the 
place  of  multiplication;  and,  in  the  tables  of  logarithmic 
sines,  tangents,  <fec.,  the  radius  is  10.  (Art.  103.)  There- 
fore, in  these  tables, 

cos-f  sec=20.     Or  sec=20 — cos. 
Again,  by  Art  93,  proportion  6, 

sinXcosec=Ra. 
Therefore,  in  the  tables, 
sin+cosec=20.     Or,  cosec=20 — sin.     Hence, 

112.  To  obtain  the  secant,  subtract  the  cosine  from  20; 
and  to  obtain  the  cosecant,  subtract  the  sine  from  20. 

These  subtractions  are  most  easily  performed,  by  taking 
the  right  hand  figure  from  10,  and  the  others  from  9,  as  in 
finding  the  arithmetical  complement  of  a  logarithm ;  (Art. 
55.)  observing,  however,  to  add  10  to  the  index  of  the  secant 
or  cosecant.  In  fact  the  secant  is  the  arithmetical  comple- 
ment of  the  cosine,  with  10  added  to  the  index. 

For  the  secant  =20 — cos. 

And  the  arith.  comp.  of  cos  =10 — cos.  (Art.  54.) 

So  also  the  cosecant  is  the  arithmetical  complement  of  the 
sine,  with  10  added  to  the  index.  The  tables  of  secants  and 
cosecants  are,  therefore,  of  use,  in  furnishing  the  arithmetical 
complement  of  the  sine  and  cosine,  in  the  following  simple 
manner : 


RIGHT   ANGLED   TRIANGLES.  07 

113.  For  the  arithmetical  complement  of  the  sine,  sub- 
tract 10  from  the  index  of  the  cosecant;  and  for  the  arith- 
metical complement  of  the  cosine,  subtract  10  from  the  index 
of  the  secant. 

By  this,  we  may  save  the  trouble  of  taking  each  of  the 
figures  from  9. 


SECTION  III. 

SOLUTIONS    OF    RIGHT    ANGLED    TRIANGLES. 

ART.  114.  In  a  triangle  there  are  six  parts,  three  sides, 
and  three  angles.  In  every  trigonometrical  calculation,  it  is 
necessary  that  some  of  these  should  be  known,  to  enable  us 
to  find  the  others.  TJie  number  of  parts  which  must  be 
given,  is  THREE,  one  of  which  must  be  a  SIDE. 

If  only  two  parts  be  given,  they  will  be  either  two  sides, 
a  side  and  an  angle,  or  two  angles ;  Cither  of  which  will 
limit  the  triangle  to  a  particular  form  and  size. 

If  two  sides  only  be  given,  they  may  make  any  angle  with 
each  other ;  and  may,  there- 
fore, be  the  sides  of  a  thousand 
different  triangles.  Thus,  the 
two  lines  a  and  b  may  belong 
either  to  the  triangle  ABC,  or 
ABC',  or  ABC".  So  that  it 

will  be  impossible,  from  know-     * 

ing  two  of  the  sides  of  a  trian- 
gle, to    determine  the  other  parts. 


68  BIGHT   ANGLED    TRIANGLES. 

Or,  if  a  side  and  an  angle 
only  be  given,  the  triangle 
•will  be  indeterminate.  Thus, 
if  the  side  AB  and  the  an- 
gle at  A  be  given;  they 
may  be  parts  either  of  the 
triangle  ABC,  or  ABC',  or 
ABC". 

Lastly,  if  two  angles,  or  even  if  all  the  angles  be  given, 
they  will  not  determine  the  length  of  the  sides.  "For  the  tri- 
angles ABC,  A'B'C',  A"B"C", 
and  a  hundred  others  which 
might  be  drawn,  with  sides  par- 
allel to  these,  will  all  have  the 
same  angles.  So  that  one  of 
the  parts  given  must  always  be 
a  side.  If  this  and  any  other 
two  parts,  either  sides  or  angles,  be  known,  the  other  three 
may  be  found,  as  will  be  shown,  in  this  and  the  following 
section. 

115.  Triangles  are  either  right  angled  or  oblique  angled. 
The  calculations  of  the  former  are  the  most  simple,  and  those 
which  we  have  the  most  frequent  occasion  to  make.    A  great 
portion  of  the  problems  in  the  mensuration  of  heights  and 
distances,  in  surveying,  navigation  and  astronomy,  are  solved 
by  rectangular  trigonometry.     Any  triangle  whatever  may 
be  divided  into  two  right  angled  triangles,  by  drawing  a  per- 
pendicular from  one  of  the  angles  to  the  opposite  side. 

116.  One  of  the  six  parts  in  aright  angled  triangle,  is 
always  given,  viz.  the  right  angle.     This  is  a  constant  quan- 
tity ;  while  the  other  angles  and  the  sides  are  variable.     It 
is  also  to  be  observed,  that,  if  one  of  the  acute  angles  is 
given,  the  other  is  known  of  course.     For  one  is  the  com- 
plement of  the  other.  (Art.  76,  77.)  So  that,  in  a  right  angled 
triangle,  subtracting  one  of  the  acute  angles  from  90°  gives  the 


RIGHT    ANGLED  TRIANGLES. 


69 


other.  There  remain,  then,  only  four  parts,  one  of  the  acute 
angles,  and  the  three  sides,  to  be  sought  by  calculation.  If 
any  two  of  these  be  given,  with  the  right  angle,  the  others 
may  be  found. 

117.  To  illustrate  the  method  of 
calculation,  let  a  case  be  supposed  in 
which  a  right  angled  triangle  CAD, 
has  one  of  its  sides  equal  to  the 
radius  to  which  the  trigonometrical 
tables  are  adapted. 

In  the  first  place,  let  the  base  of  the  p  . 
triangle  be  equal  to  the  tabular  radius.  Then,  if  a  circle  be  de- 
scribed, with  this  radius,  about  the  angle  C  as  a  centre,  DA 
will  be  the  tangent,  and  DC  the  secant  of  that  angle.  (Art. 
84,  85.)  So  that  the  radius,  the  tangent,  and  the  secant  of 
the  angle  at  C,  constitute  the  three  sides  of  the  triangle. 
The  tangent,  taken  from  the  tables  of  natural  sines,  tangents, 
<fec.,  will  be  the  length  of  the  perpendicular  ;  and  the  secant 
will  be  the  length  of  the  hypothenuse.  If  the  tables  used  be 
logarithmic,  they  will  give  the  logarithms  of  the  lengths  of 
the  two  sides. 

In  the  same  manner,  any 
right  angled  triangle  whatever, 
whose  base  is  equal  to  the 
radius  of  the  tables,  will  have 
its  other  two  sides  found 
among  the  tangents  and  se- 
cants. Thus,  if  the  quadrant 
AH,  be  divided  into  portions 
of  15°  each ;  then,  in  the 
triangle 

CAD,  AD  will  be  the  tan,  and  CD  the  sec  of  15°, 
In  CAD/,  AD'  will  be  the  tan,  and  CD7  the  sec  of  30°, 
In  CAD",  AD"  will  be  the  tan,  and  CD"  the  sec  of  45°,<fec. 


70 


RIGHT   ANGLED    TRIANGLES. 


118.  In  the    next    place,   let 
the  hypothenuse  of  a  right  angled 
triangle  CBF,    be   equal   to   the 
radius  of  the  tables.     Then,  if  a 
circle  be  described,  with  the  given 
radius,  and  about  the  angle  C  as 
a   centre;    BF  will   be   the  sine, 
and  BC  the  cosine  of  that  angle. 
(Art.     82,    89.)      Therefore    the 

sine  of  the  angle  at  C,  taken  from  the  tables,  will  be  the 
length  of  the  perpendicular,  and  the  cosine  will  be  the  length 
of  the  base. 

And  any  right  angled  triangle 
whatever,  whose  hypothenuse 
is  equal  to  the  tabular  radius, 
will  have  its  other  two  sides 
found  among  the  sines  and  co- 
sines. Thus,  if  the  quadrant 
AH,  be  divided  into  portions  of 
15°  each  in  the  points  F,  F',  F", 
&c. ;  then,  in  the  triangle, 

CBF,  FB  will  be  the  sin,  and  CB  the  cos,  of  15°, 
In  CB'F',  F'B'  will  be  the  sin,  and  CB'  the  cos,  of  30°, 
In  CB"F",  F"B"  will  be  the  sin,  and  CB"  the  cos,  of  45°,  &c, 

119.  By  merely  turning  to  the  tables,  then,  we  may  find 
the  parts  of  any  right  angled  triangle  which  has  one  of  its 
sides  equal  to  the  radius  of  the  tables.     But  for  determin- 
ing the  parts  of  triangles  which  have  not  any  of  their  sides 
equal  to  the  tabular  radius,  the  following  proportion  is  used ; 

As  the  radius  of  one  circle, 
To  the  radius  of  any  other  ; 
So  is  a  sine,  tangent,  or  secant,  in  one, 
To  the  sine,  tangent,  or  secant,  of  the  same  number 
of  degrees,  in  the  other. 


RIGHT   ANGLED    TRIANGLES. 


71 


In  the  two  concentric 
circles  AHM,  ahm,  the 
arcs  AG  and  ag,  contain 
the  same  number  of  de- 
grees. (Art.  74.)  The 
sines  of  these  arcs  are 
BG  and  bg,  the  tangents 
AD  and  ad,  and  the  se- 
cants CD  and  Cd.  The  four  triangles,  CAD,  CBG,  Cad, 
and  Cbg,  are  similar.  For  each  of  them,  from  the  nature 
of  sines  and  tangents,  contains  one  right  angle  ;  the  angle  at 
C  is  common  to  them  all ;  and  the  other  acute  angle  in  each 
is  the  complement  of  that  at  C.  (Art.  77.)  We  have,  then, 
the  following  proportions.  (Euc.  4.  6.*) 

1.  CG  :  Cg  :  :  BG  :  bg. 

That  is,  one  radius  is  to  the  other,  as  one  sine  to  the  other. 

2.  CA  :  Ca  :  :  DA  :  da. 

That  is,  one  radius  is  to  the  other,  as  one  tangent  to  the  other. 

3.  CA  t  Ca  :  :  CD  ;  Cd. 

That  is,  one  radius  is  to  the  other,  as  one  secant  to  the  other. 
Cor.   BG  :  bg  :  :  DA  :  da  :  :  CD  :  Cd. 

That  is,  as  the  sine  in  one  circle,  to  the  sine  in  the  other ; 
so  is  the  tangent  in  one,  to  the  tangent  in  the  other ;  and  so 
is  the  secant  in  one,  to  the  secant  in  the  other. 

This  is  a  general  principle,  which  may  be  applied  to  most 
trigonometrical  calculations.  If  one  of  the  sides  of  the  pro- 
posed triangle  be  made  radius,  each  of  the  other  sides  will 
be  the  sine,  tangent,  or  secant,  of  an  arc  described  by  this 
radius.  Proportions  are  then  stated,  between  these  lines, 
and  the  tabular  radius,  sine,  tangent,  <fec. 


*  Thomson  18.  4. 


RIGHT   ANGLED    TRIANGLES. 


120.  A  line  is  said  to  be  made  radius,  when  a  circle  is 
described,  or  supposed  to  be  described,  whose  semi-diameter 
is  equal  to  the  line,  and  whose  centre  is  at  one  end  of  it. 

121.  In  any  right  angled  triangle,  if  the  HYPOTHENUSE  be 
made  radius,  one  of  the  leys  will  be  a  SINE  of  its  opposite 
angle,  and  the  other  leg  a  COSINE  of  the  same  angle. 

Thus,  if  to  the  triangle  ABC 
a  circle  be  applied  whose  radius 
is  AC,  and  whose  centre  is  A, 
then  BC  will  be  the  sine,  and 
BA  the  cosine,  of  the  angle  at 
A.  (Art.  82,  89.) 

If,  while  the  same  line  is 
radius,  the  other  end  C  be  made 

the  centre,  then  BA  will  be  the  sine,  and  BC  the  cosine,  of 
the  angle  at  C. 

122.  If  either  of  the  LEGS  be  made  radius,  the  other  leg  will 
be  a  TANGENT  of  its  opposite  angle,  and  the  hypothenuse  will 
be  a  SECANT  of  the  same  angle  ;  that  is,  of  the  angle  between 
the  secant  and  the  radius. 


Rod 


Thus,  if  the  base  AB  (Fig.  15.)  be  made  radius,  the  centre 
being  at  A,  BC  will  be  the  tangent,  and  AC  the  secant,  of 
the  angle  at  A.  (Art.  84.  85.) 

But,  if  the  perpendicular  BC,  (Fig.  16.)  be  made  radius, 
with  the  centre  at  C,  then  AB  will  be  the  tangent,  and  AC 
the  secant,  of  the  angle  at  C. 

123.  As  the  side  which  is  the  sine,  tangent,  or  secant 
of  one  of  the  acute  angles,  is  the  cosine,  cotangent,  or  cose- 


RIGHT    ANGLED    TRIANGLES.  73 

cant  of  the  other  ;  (Art.  89,)  the  perpendicular  BC  (Fig.  14.) 
is  the  sine  of  the  angle  A,  and  the  cosine  of  the  angle  C ; 
while  the  base  AB,  is  the  sine  of  the  angle  C,  and  the  cosine 
of  the  angle  A. 

If  the  base  is  made  radius,  as  in  Fig  15,  the  perpendicular 
BC  is  the  tangent  of  the  angle  A,  and  the  cotangent  of  the 
angle  C  ;  while  the  hypothenuse  is  the  secant  of  the  angle 
A,  and  the  cosecant  of  the  angle  C. 

If  the  perpendicular  is  made  radius,  as  in  Fig.  16,  the 
base  AB  is  the  tangent  of  the  angle  C,  and  the  cotangent  of 
the  angle  A ;  while  the  hypothenuse  is  the  secant  of  the  angle 
C,  and  the  cosecant  of  the  angle  A. 

124.  Whenever  a  right  angled  triangle  is  proposed,  whose 
sides  or  angles  are  required  ;  a  similar  triangle  may  be 
formed,  from  the  sines,  tangents,  <fec.,  of  the  tables.  (Art. 
117,  118.)  The  parts  required  are  then  found,  by  stating 
proportions  between  the  similar 
sides  of  the  two  triangles.  If 
the  triangle  proposed  be  ABC, 
(Fig.  17.)  another  abc  may  be 
formed,  having  the  same  angles 
with  the  first,  but  differing  from 
it  in  the  length  of  its  sides,  so  as 
to  correspond  with  the  numbers  in  the  tables.  If  similar  sides 
be  made  radius  in  both,  the  remaining  similar  sides  will  be 
lines  of  the  same  name  ;  that  is,  if  the  perpendicular  in  one 
of  the  triangles  be  a  sine,  the  perpendicular  in  the  other  will 
be  a  sine ;  if  the  base  in  one  be  a  cosine,  the  base  in  the 
other  will  be  a  cosine,  &c. 

If  the  hypothenuse  in  each  triangle  be  made  radius,  as  in 
Fig.  14,  the  perpendicular  be,  will  be  the  tabular  sine  of  the 
angle  at  a  ;  and  the  perpendicular  BC,  will  be  a  sine  of 
the  equal  angle  A,  in  a  circle  of  which  AC  is  radius. 

If  the  base  in  each  triangle  be  made  radius,  as  in  Fig.  15, 
then  the  perpendicular  be,  will  be  the  tabular  tangent  of  the 

7 


74  RIGHT   ANGLED   TRIANGLES. 

angle  at  a  ;  and  BC  -will  be  a  tangent  of  the  equal  angle  A, 
in  a  circle  of  which  AB,  is  radius,  &e. 

125.  From  the  relations  of  the  similar  sides  of  these  tri- 
angles, are  derived  the  two  following  theorems,  which  are 
sufficient  for  calculating  the  parts  of  any  right  angled  tri- 
angle whatever,  when  the  requisite  data  are  furnished.  One 
is  used,  when  a  side  is  to  be  found ;  the  other,  when  an 
angle  is  to  be  found. 


THEOREM  I. 
126.  When  a  side  is  required ; 

As    THE    TABULAR    SINE,    TANGKNT,    &<X,    OF    THE 
SAME    NAME    WITH    THE    GIVEN    SIDE, 

To  THE  GIVEN  SIDE; 

SO    IS    THE    TABULAR    SINE,  TANGENT,  <feC.,  OF  THE 

SAME    NAME    WITH    THE    REQUIRED   SIDE, 
To  THE   REQUIRED    SIDE. 

It  will  be  readily  seen,  that  this  is  nothing  more  than  a 
statement  in  general  terms,  of  the  proportions  between  the 
similar  sides  of  two  triangles,  one  proposed  for  solution,  and 
the  other  formed  from  the  numbers  in  the  tables. 

Thus,  if  the  hypothenuse  be 
given,  and  the  base  or  perpen- 
dicular be  required ;  then  in 
Fig.  14,  where  ac  is  the  tabular 
radius,  be  the  tabular  sine  of  a, 
or  its  equal  A,  and  ab  the  tab- 
ular sine  of  C;  (Art.  124.) 

ac  :  AC  :  :  be  :  BC,  that  is,  B  :  AC  :  :  sin  A  :  BC. 
«c  ;  AC  : :  o&  :  AB,  B  :  AC  : :  sin  C  ;  AB. 


RIGHT   ANGLED   TRIANGLES. 


15 


In  Fig.  15,  where  ab  is  the  tabular  radius,  ac  the  tabular 
secant  of  A,  and  be  the  tabular  tangent  of  A  ; 
ac  :  AC  :  :  be  :  BC,  that  is,  sec  A  :  AC  :  :  tan  A  :  BC. 
ac  :  AC  :  :  ab  :  AB,  sec  A  :  AC  :  :  R  ;  AB. 

In  Fig.  16,  where  be  is  the  tabular  radius,  ac  the  tabular 
secant  of  C,  and  ab  the  tabular  tangent  of  C  ; 
ac  :  AC  :  :  be  :  BC,  that  is,  sec  C  :  AC  :  :  R  :  BC. 
ac  :  AC  :  :  ab  :  AB,  sec  C  :  AC  :  :  tan  C  ;  AB. 

THEOREM  II. 
127.  When  an  angle  is  required ; 

As    THE    GIVEN    SIDE    MADE    RADIUS, 
To    THE    TABULAR    RADIUS  J 
So    IS    ANOTHER    GIVEN    SIDE, 

TO    THE    TABULAR    SINE,    TANGENT,    AC.,    OF     THE 
SAME    NAME. 

Thus,  if  the  side  made  radius,  and  one  other  side  be  given, 
then,  in  Fig.  14, 

AC  :  ac  :  :  BC  :  be,  that  is,  AC 
AC  :  ac  :  :  AB  :  ab,  AC 

In  Fig.  15, 

AB  :  ab  :  :  BC  :  be,  that  is,  AB 
AB  :  ab  :  :  AC  :  ac,  AB 

In  Fig.  16, 
AB  :  ab,  that  is,  BC 


BC  :  be 
BC  ;  be 


:  AC  :  ac, 


BO  ; 


R: 
R: 

BC 
AB 

sin  A. 
sin  C. 

R: 
R: 

BC 
AC 

tan  A. 
sec  A. 

R: 

AB 
AC 

tanC. 
sec  C. 

76  RIGHT   ANGLED    TRIANGLES. 

It  will  be  observed  that  in  these  theorems,  angles  are  not 
introduced,  though  they  are  among  the  quantities  which  are 
either  given  or  required,  in  the  calculation  of  triangles.  But 
the  tabular  sines,  tangents,  &c.,  may  be  considered  the  repre- 
sentatives of  angles,  as  one  may  be  found  from  the  other,  by 
merely  turning  to  the  tables. 

128.  In  the  theorem  for  finding  a  side,  the  first  term  of 
the  proportion  is  a  tabular  number.     But,  in  the  theorem  foi 
finding  an  angle,  the  first  term  is  a  side.     Hence,  in  apply- 
ing  the  proportions  to  particular  cases,  this  rule  is  to  be  ob- 
served ; 

To  find  a  SIDE,  begin  with  a  tabular  number, 
To  find  an  ANGLE,  begin  with  a  side. 

Radius  is  to  be  reckoned  among  the  tabular  numbers. 

129.  In  the  theorem  for  finding  an  angle,  the  first  term  is 
a  side  made  radius.     As  in  every  proportion,  the  three  first 
terms  must  be  given  to  enable  us  to  find  the  fourth,  it  is  evi- 
dent,  that  where  this  theorem  is  applied,  the  side  made 
radius  must  be  a  given  one.     But,  in  the  theorem  for  finding 
a  side,  it  is  not  necessary  that  either  of  the  terms  should  be 
radius.  Hence, 

130.  To  find  a  SIDE,  ANY  side  may  be  made  radius. 

To  find  an  ANGLE,  a  GIVEN  side  must  be  made  radius. 

It  will  generally  be  expedient,  in  both  cases,  to  make 
radius  one  of  the  terms  in  the  proportion ;  because,  in  the 
tables  of  natural  sines,  tangents,  &c.,  radius  is  1,  and  in  the 
logarithmic  tables  it  is  10.  (Art.  103.) 

131.  The  proportions  in  Trigonometry  are  of  the  same 
nature  as  other  simple  proportions.    The  fourth  term  is  found, 
therefore,  as  in  the  Rule  of  Three  in  Arithmetic,  by  multi- 
plying together  the  second  and  third  terms,  and  dividing  their 
product  by  the  first  term.     This  is  the  mode  of  calculation, 
when  the  tables  of  natural  sines,  tangents,  <fec.,  are  used. 
But  the  operation  by  logarithms  is  so  much  more  expeditious, 


RIGHT   ANGLED    TRIANGLES.  77 

that  it  has  almost  entirely  superseded  the  other  method.  In 
logarithmic  calculations,  addition  takes  the  place  of  multipli- 
cation ;  and  subtraction  the  place  of  division. 

The  logarithms  expressing  the  lengths  of  the  sides  of  a 
triangle,  are  to  be  taken  from  the  tables  of  common  loga- 
rithms. The  logarithms  of  the  sines,  tangents,  &c.t  are  found 
in  the  tables  of  artificial  sines,  &c.  The  calculation  is  then 
made  by  adding  the  second  and  third  terms,  and  subtracting 
the  first.  (Art.  52.) 

132.  The  logarithmic  radius  10,  or,  as  it  is  written  in  the 
tables,  10.00000,  is  so  easily  added  and  subtracted,  that  the 
three  terms  of  which  it  is  one,  may  be  considered  as,  in 
effect,  reduced  to  two.     Thus,  if  the  tabular  radius  is  in  the 
first  term,  we  have  only  to  add  the  other  two  terms,  and 
then  take  10  from  the  index ;  for  this  is  subtracting  the  first 
term.     If  radius  occurs  in  the  second  term,  the  first  is  to  be 
subtracted  from  the  third,  after  its  index  is  increased  by  10. 
In  the  same  manner,  if  radius  is  in  the  third  term,  the  first  is 
to  be  subtracted  from  the  second. 

133.  Every  species   of  right   angled  triangles  may  be 
solved  upon  the  principle,  that  the  sides  of  similar  triangles 
are  proportional,  according  to  the  two  theorems  mentioned 
above.     There  will  be  some  advantages,  however,  in  giving 
the  examples  in  distinct  classes. 

There  must  be  given,  in  a  right  angled  triangle,  two  of  the 
parts,  besides  the  right  angle.  (Art.  116.)  These  may  be; 

1.  The  hypothenuse  and  an  angle;  or 

2.  The  hypothenuse  and  a  leg  ;  or 

3.  A  leg  and  an  angle  ;  or 

4.  The  two  legs. 


CASK  I. 

thenuse, 
angle,       }  "      ""  j  Perpendicular. 


134.  Given  i  Th«  hypothenuse,  )  tofind  (  The  base  and 
(  And  an  angle,       )  ( 


78 


RIGHT   ANGLED    TRIANGLES. 


\C 


\ 


Ex.  1.  If  the  hypothenuse  AC,* 
be  45  miles,  and  the  angle  at  A 
32°  20',  what  is  the  length  of  the 
base  AB,  and  the  perpendicular 
BC? 

In  this  case,  as  sides  only  are 
required,  any  side  may  be  made 
radius.  (Art.  130.) 

If  the  hypothenuse  be  made 
radius,  BC  will  be  the  sine  of 
A,  and  AB  the  sine  of  C,  or  the 
cosine  of  A.  (Art.  121.)  And 
if  abc  be  a  similar  triangle, 
whose  hypothenuse  is  equal  to 
the  tabular  radius,  be  will  be  the 
tabular  sine  of  A,  and  db  the 
tabular  sine  of  C.  (Art.  124.) 

To  find  the  perpendicular,  then,  by  Theorem  I,  we  have 
this  proportion ; 

ac  :  AC  :  :  be  :  BC. 
Or  R  :  AC  :  :  Sin  A  :  BC. 

Whenever  the  terms  Radius,  Sine,  Tangent,  &c.,  occur  in  a 
proportion  like  this,  the  tabular  Radius,  &c.,  is  to  be  under- 
stood, as  in  Arts.  126,  127. 

The  numerical  calculation,  to  find  the  length  of  BC,  may 
be  made,  either  by  natural  sines,  or  by  logarithms.  See 
Art  131. 

By  natural  Sines. 
1  t  45  :  :  0.53484  :  24.068=BC. 

*  The  parts  which  are  given  are  distinguished  by  a  mark  across  the 
line,  or  at  the  opening  of  the  angle,  and  the  parts  required  by  a  cipher. 


RIGHT    ANGLED    TRIANGLES. 


Computation  by  Logarithms, 
As  radius  10.00000 

To  the  hypothenuse         45  1.65321 

So  is  the  Sine  of  A          32°  20'          9.72823 
To  the  perpendicular       24.068  1.38144 

Here  the  logarithms  of  the  second  and  third  terms  are 
added,  and  from  the  sum,  the  first  term  10  is  subtracted, 
(Art  132.)  The  remainder  is  the  logarithm  of  24.068=BC. 
•  Subtracting  the  angle  at  A  from  90°,  we  have  the  angle  at 
C=^57°  40'.  (Art.  116.)  Then  to  find  the  base  AB; 

ac  :  AC  :  :  ab  :  AB 
Or  R  ;  AC  :  :  Sin  C  :  AB=38.023. 

Both  the  sides  required  are  now  found,  by  making  the 
hypothenuse  radius.  The  results  here  obtained  may  be  veri- 
fied, by  making  either  of  the  other  sides  radius. 


If  the  base  be  made  radius,  as  in  Fig.  15,  the  perpen- 
dicular will  be  the  tangent,  and  the  hypothenuse  the  secant 
of  the  angle  at  A.  (Art  122.)  Then, 

Sec  A  :  AC  :  :  R  :  AB 

R  I  AB  :  :  Tan  A  :  BC 

By  making  the  arithmetical  calculations,  in  these  two  pro- 
portions, the  values  of  AB  and  BC,  will  be  found  the  same 
as  before. 

If  the  perpendicular  be  made  radius,  as  in  Fig.  16,  AB 
will  be  the  tangent,  and  AC  the  secant  of  the  angle  at  C. 
Then, 


80 


RIGHT   ANGLED    TRIANGLES. 


Sec  C  :  AC  :  :  R  :  BC 
R  ;  BC  :  :  Tan  C  :  AB 

Ex.  2.  If  the  hypothennse  of  a  right  angled  triangle  be 
250  rods,  and  the  angle  at  the  base  46°  30' ;  what  is  the 
length  of  the  base  and  perpendicular  ? 

Ans.  The  base  is  172.1  rods,  and  the  perpendic.  181.35. 


CASE  II. 


135.  Given  \  The  hyP°<*enuse,  j   to  find  j  The  angles  and 
(  And  one  leg.  )  (  The  other  leg. 


18 


one  leg.  )  (  The  other  leg. 

Ex.  1.  If  the  hypothenuse  be  35 
leagues,  and  the  base  26 ;  what  is 
the  length  of  the  perpendicular* 
and  the  quantity  of  each  of  the 
acute  angles  ? 

To  find  the  angles  it  is  necessary 
that  one  of  the  given  sides  be  made 
radius.  (Art.  130.) 

If  the  hypothenuse  be  radius,  the  base  and  perpendicular 
will  be  sines  of  their  opposite  angles.     Then, 

R  :  :  AB  :  Sin  C=47°  58i'     * 


AC 


And  to  find  the  perpendicular  by  Theorem  I ; 
R  :  AC  :  :  Sin  A  :  BC=23.43 

If  the  base  be  radius,  the  perpendicular  will  be  tangent, 
and  the  hypothenuse  secant  of  the  angle  at  A.  Then, 

AB  :  R  :  :  AC  :  Sec  A 
R  :  AB  :  :  Tan  A  :  BC 

In  this  example,  where  the  hypothenuse  and  base  are 
given,  the  angles  cannot  be  found  by  making  the  perpen- 
dicular radius.  For  to  find  an  angle,  a  given  side  must  be 
made  radius.  (Art.  130.) 


RIGHT   ANGLED   TRIANGLES. 


81 


136,  Ex.  2.  If  the  hypothenuse  be  54 
miles,  and  the  perpendicular  48  miles, 
what  are  the  angles,  and  the  base  ? 

Making  the  hypothenuse  radius. 

AC  :  R  :  :  BC  :  Sin  A 
R  I  AC  :  :  Sin  C  :  AB 


The  numerical  calculation  will  give  A=62°  44'  and  AB 
-=24.74. 

Making  the  perpendicular  radius. 

BC  :  R  :  :  AC  :  Sec  C 
R  :  BC  :  :  Tan  C  :  AB 

The  angles  cannot  be  found  by  making  the  base  radius, 
when  its  length  is  not  given. 


CASE  III. 


137.  Given   $  TKanSles>  1  to  find    j  The  h7P°tlien™e, 
(  and  one  leg.  )  (  and  the  other  leer. 


Ex.  1.  If  the  base  be  60,  and  the 
angle  at  the  base  47°  12',  what  is  the 
length  of  the  hypothenuse  and  the  per- 
pendicular ? 

In  this  case,  as  sides  only  are  re- 
quired, any  side  may  be  radius.  A 


Making  the  hypothenuse  radius. 

Sin  C  :  AB  :  :  R  :  AC=88.31 
R  t  AC  :  :  Sin  A  :  BC«=64.8 


82  RIGHT  ANGLED   TRIANGLES, 

Making  the  base  radius,    (Fig.  20.) 
R  :  AB  :  :  Sec  A  :  AC 
R  :  AB  :  :  Tan  A  :  BO 

Making  the  perpendicular  radius. 
Tan  C  :  AB  :  :  R  :  BO 
R  :  BC  :  :  Sec  G  :  AC 


138.  Ex.  2.  If  the  perpen- 
dicular be  74,  and  the  angle 
C  61°  27',  what  is  the  length 
of  the  base  and  the  hypothe- 
nuse  ? 


Making  the  hypothenuse  radius. 
Sin  A  :  BC  :  :  R  :  AC 
R  t  AC  :  :  sin  C  :  AB 

Making  the  base  radius. 

Tan  A  :  BC  :  :  R  :  AB 
R  t  AB  :  :  sec  A  :  AC 

Making  the  perpendicular  radius. 
R  t  BC  :  :  sec  C  :  AC 
R  t  BC  :  :  tan  C  :  AB 
The  hypothenuse  is  154.83  and  the  base  136. 

CASE  IV. 

139.  Given  }  ™e  ba*e>  "*  {  to  find  j  ™e  hypothenuse, 
(  Perpendicular,  )  (  And  the  angles. 

Ex.  1.  If  the  base  be  284,  and 
the  perpendicular  192,  what 
are  the  angles,  and  the  hypothe- 
nuse? 

In  this  case,  one  of  the  legs 


RIGHT  ANGLED    TRIANGLES.  83 

must  be  made  radius,  to  find  an  angle ;  because  the  hypothe- 
nuse  is  not  given. 

Making  the  base  radius. 

AB  ;  R  :  :  BC  :  tan  A=34°  47 
R  ;  AB  :  :  sec  A  :  AC=342.84 

Making  the  perpendicular  radius. 

BC  :  R  :  :  AB  :  tan  0 
R  :  BC  :  :  sec  C  :  AC 

Ex.  2.  If  the  base  be  640,  and  the  perpendicular  480, 
what  are  the  angles  and  hypothenuse  ? 

Ans.  The  hypothenuse  is  800,  and  the  angle  at  the  base 
36°  52'  12". 

Examples  for  Practice. 

1.  Given  the  hypothenuse  68,  and  the  angle  at  the  base 

89°  17' ;  to  find  the  base  and  perpendicular. 

2.  Given  the  hypothenuse  850,  and  the  base  594,  to  find 

the  angles,  and  the  perpendicular. 

3.  Given  the  hypothenuse  78,  and  perpendicular  57,  to 

find  the  base,  and  the  angles. 

4.  Given  the  base  723,  and  the  angle  at  the  base  64°  18', 

to  find  the  hypothenuse  and  perpendicular. 

5.  Given  the  perpendicular  632,  and  the  angle  at  the  base 

81°  36',  to  find  the  hypothenuse  and  the  base. 

6.  Given  the  base  32,  and  the  perpendicular  24,  to  find 

the  hypothenuse,  and  the  angles, 

140.  The  preceding  solutions  are  all  effected,  by  means 
of  the  tabular  sines,  tangents,  and  secants.  But,  when  any 
two  sides  of  a  right  angled  triangle  are  given,  the  third  side 
may  be  found,  without  the  aid  of  the  trigonometrical  tables, 
by  the  proposition,  that  the  square  of  the  hypothenuse  is  equal 


84  RIGHT   ANGLED   TRIANGLES. 

to   the   sum  of  the  squares  of  the  two  perpendicular  sides. 
(Euo.  47.  1.) 

If  the  legs  be  given,  extracting  the  square  root  of  the  sum 
of  their  squares,  will  give  the  hypothenuse.  Or,  if  the  hypo- 
thenuse  and  one  leg  be  given,  extracting  the  square  root  of 
the  difference  of  the  squares,  will  give  the  other  leg. 

Let  A=the  hypothenuse       \ 

j»=the  perpendicular    >  of  a  right  angled  triangle. 
6==the  base  ) 


Then         #=&2-fy,  or  (Alg.  248.) 


By  trans.  62=A3—  p*,  or  6=vA2—  ^ 

And         p*=h*  —  62,  or 


Ex.  1.  If  the  base  is  32,  and  the  perpendicular  24,  what 
is  the  hypothenuse  ?  Ans.  40. 

2.  If  the  hypothenuse  is  100,  and  the  base  80,  what  is 
the  perpendicular  ?  Ans.  60. 

3.  If  the  hypothenuse  is  300,  and  the  perpendicular  220, 
what  is  the  base  ? 

Ans.  3002  —  2202=4160,  the  root  of  which  is  204  nearly. 

141.  It  is  generally  most  convenient  to  find  the  difference 
of  the  squares  by  logarithms.  But  this  is  not  to  be  done  by 
subtraction.  For  subtraction,  in  logarithms,  performs  the 
office  of  division.  (Art.  41.)  If  we  subtract  the  logarithm 
of  6a  from  the  logarithm  of  A2,  we  shall  have  the  logarithm, 
not  of  the  difference  of  the  squares,  but  of  their  quotient. 
There  is,  however,  an  indirect,  though  very  simple  method, 
by  which  the  difference  of  the  squares  may  be  obtained  by 
logarithms.  It  depends  on  the  principle,  that  the  difference 
of  the  squares  of  two  quantities  is  equal  to  the  product  of  the 
sum  and  difference  of  the  quantities.  (Alg.  191.)  Thus, 


OBLIQUE    ANGLED    TRIANGLES.  85 

as  will  be  seen  at  once,  by  performing  the  multiplication.  The 
two  factors  may  be  multiplied  by  adding  their  logarithms. 
Hence, 

142.  To  obtain  the  difference  of  the  squares  of  two  quanti~ 
ties,  add  the  logarithm  of  the  sum  of  the  quantities  to  the 
logarithm  of  their  difference.  After  the  logarithm  of  the 
difference  of  the  squares  is  found  ;  the  square  root  of  this 
difference  is  obtained,  by  dividing  the  logarithm  by  2. 
(Art.  47.) 

Ex.  1.  If  the  hypothenuse  be  75  inches,  and  the  base  45, 
what  is  the  length  of  the  perpendicular  ? 

Sum  of  the  given  sides      120  log.  2.07918 

Difference  of       do.              30  1.47712 

Dividing  by  2)3.55630 

Side  required                        60  1.77815 

2.  If  the  hypothenuse  is  135,  and  the  perpendicular  108, 
what  is  the  length  of  the  base  ?  Ans.  81. 


SECTION  IV. 

SOLUTIONS    OF    OBLiaUE    ANGLED    TRIANGLES. 

ART.  143.  The  sides  and  angles  of  oblique  angled  trian- 
gles may  be  calculated  by  the  following  theorems. 

THEOREM  I. 
In  any  plane  triangle,  THE  SINES  or  THE  ANGLES  ARE  AS 

THEIR   OPPOSITE    SIDES. 

8 


86  OBLIQUE   ANGLED   TRIANGLES. 

Let  the  angles  be  denoted  by  the  letters  A,  B,  C,  and  their 
opposite  sides  by  a,  6,  c,  as  in  Fig.  23  and  24.     From  one 


of  the  angles,  let  the  line  p  be  drawn  perpendicular  to  the 
opposite  side.  This  will  fall  either  within  or  without  the 
triangle. 

1.  Let  it  fall  within  as  in  Fig.  23.     Then,  in  the  right 
angled  triangles  ACD,  and  BCD,  according  to  Art.  126, 

R  :  b  :  :  sin  A  :  p 
R  t  a  :  :  sin  B  ;  p 

Here,  the  two  extremes  are  the  same  in  both  proportions. 
The  other  four  terms  are,  therefore,  reciprocally  proportion- 
al :*  that  is, 

a  :  b  :  :  sin  A  t  sin  B. 

2.  Let  the  perpendicular^?  fall  without  the  triangle,  as  in 
Fig.  24.     Then  in   the  right  angled  triangles   ACD  and 
BCD; 

E  :  5  :  :  sin  A  :  p 
R  t  «  :  *  sin  B  t  p 
Therefore,  as  before, 

a  \  b  :  :  sin  A  ;  sin  B. 

Sin  A  is  here  put  both  for  the  sine  of  DAC,  and  for  that 
of  BAG.     For,  as  one  of  these  angles  is  the  supplement  of 
the  other,  they  have  the  same  sine.  (Art.  90.) 
i    The  sines  which  are  mentioned  here,  and  which  are  used 


*  Euclid,  23.  6, 


OBLIQUE    ANGLED   TRIANGLES.  67 

in  calculation  are  tabular  sines.  But  the  proportion  will  be 
the  same,  if  the  sines  be  adapted  to  any  other  radius.  (Art. 
119.) 

THEOREM  II. 
144.  In  a  plane  triangle, 

As    THE    SUM    OP    ANY   TWO    OF    THE    SIDES, 

To    THEIR   DIFFERENCE  ; 

So     IS     THE     TANGENT     OF    HALF    THE    SUM    OF   THE 

OPPOSITE    ANGLES  J 
To    THE    TANGENT    OF   HALF   THEIR   DIFFERENCE. 

Thus,  the  sum 

of  AB  and  AC,  A c 

is  to  their  differ- 
ence ;  as  the  tan- 
gent of  half  the 
sum  of  the  an- 
gles ACB  and 
ABC,  to  the  tan- 
gent of  half  their  difference. 

Demonstration. 

Extend  CA  to  G,  making' AG  equal  to  AB;  then  CG  is 
the  sum  of  the  two  sides  AB  and  AC.  On  AB,  set  off  AD, 
equal  to  AC  ;  then  BD  is  the  difference  of  the  sides  AB  and 
AC. 

The  sum  of  the  two  angles  ACB  and  ABC,  is  equal  to 
the  sum  of  ACD  and  ADC  ;  because  each  of  these  sums  is 
the  supplement  of  CAD.  (Art.  79.)  But  as  AC=»=AD  by 
construction,  the  angle  ADC=ACD  (Euc.  5  1.*)  There- 
fore ACD  is  half  the  sum  of  ACB  and  ABC.  As  AB=AG, 
the  angle  AGB=ABG,  or  DBE.  Also,  GCE,  or  ACD=- 
ADC=BDE.  (Euc.  15.  l.f)  Therefore  in  the  triangles 

*  Thomson's  Legendre,  11.  1.  f  rbid-  *•  *• 


88 


OBLIQUE    ANGLED    TRIANGLES. 


GCE,  and  DBE, 
the  two  remain- 
ing angles  DEB, 
and  CEG,  are 
equal;  (Art.  79.) 
So  that  CE  is 
perpendicular  to 
BG.  (Euc.  Def. 
7.  1.*)  If  then  CE  is  made  radius,  GE  is  the  tangent 
of  GCE,  (Art.  84.)  that  is,  the  tangent  of  half  the  sum  of 
the  angles  opposite  to  AB  and  AC. 

If  from  the  greater  of  the  two  angles  ACB  and  ABC, 
there  be  taken  ACD  their  half  sum;  the  remaining  angle 
ECB  will  be  their  half  difference.  The  tangent  of  this  an- 
gle, CE  being  radius,  is  EB,  that  is,  the  tangent  of  half  the 
difference  of  the  angles  opposite  to  AB  and  AC.  We  have  then, 

CG=the  sum  of  the  sides  AB  and  AC  ; 

DB=their  difference ; 

GE=the  tangent  of  half  the  sum  of  the  opposite  angles  ; 

EB=the  tangent  of  half  their  difference. 

But  by  similar  triangles, 
CG  :  DB  :  :  GE  :  EB.  Q.  E.  D. 

THEOREM  III. 

145.  If  upon  the  longest  side  of  a  triangle,  a  perpendicular 
be  drawn  from  the  opposite  angle  ; 

As    THE    LONGEST   SIDE, 

To  THE  SUM  OF  THE  TWO  OTHERS  J 

So  18  THE  DIFFERENCE  OF  THE 
LATTER, 

To  THE  DIFFERENCE  OF  THE  SEG- 
MENTS MADE  BY  THE  PERPEN- 
DICULAR. 


•  Thomfon'i  Legendrt,  Def  12,  1. 


OBLIQUE    ANGLED    TRIANGLES.  89 

In  the  triangle  ABC,  if  a  perpendicular  be  drawn  from 
C  upon  AB; 

AB  ;  CB+CA  :  :  CB— CA  :  BP— PA.* 

Demonstration. 

Describe  a  circle  on  the  centre  C,  and  with  the  radius  BC. 
Through  A  and  C,  draw  the  diameter  LD,  and  extend  BA 
to  H.  Then  by  (Euc.  35.  3. f) 

ABxAH=ALxAD 
Therefore, 

AB  :  AD  :  :  AL  :  AH 

But  AD=CD+CA=CB+CA 
And  AL=CL— CA=CB— CA 
And  AH=HP— PA=BP— PA  (Euc.  3.  3.— Thorn.  6.  2.) 

If,  then,  for  the  three  last  terms  hi  the  proportion,  we  sub- 
stitute their  equals,  we  have, 

AB  :  CB+CA  :  :  CB— CA  :  PB— PA. 

146.  It  is  to  be  observed,  that  the  greater  segment  is  next 
the  greater  side.     If  BC  is  greater  than  AC,  PB  is  greater 
than  AP.     With  the  radius  AC,  describe  the  arc  AN.     The 
segment  NP=AP.  (Euc.  3.3.)   But  BP  is  greater  than  NP. 

14 7.  The  two  segments  are  to  each  other,  as  the  tangents 
of  the  opposite  angles,  or  the  cotangents  of  the  adjacent  an- 
gles.    For,  in  the  right  angled  triangles  AGP,  and  BCP,  if 
CP  be  made  radius,  (Art.  126.) 

R  t  PC  :  :  Tan  ACP  :  AP 
R  :  PC  :  :  Tan  BCP  :  BP 

Therefore,  by  equality  of  ratios,  (Alg.  346.J) 
Tan  ACP  :  AP  :  :  Tan  BCP  :  BP 

*  See  note  B.      f  Thomson's  Legendre,  28.  4.    Cor.      J  Eue.  11.  & 

8* 


90  OBLIQUE    ANGLED   TRIANGLES. 

That  is,  the  segments  are  as  the  tangents  of  the  opposite 
angles.  And  the  tangents  of  these  are  the  cotangents  of  the 
adjacent  angles  A  and  B.  (Art.  89.) 

Cor.  The  greater  segment  is  opposite  to  the  greater  angle. 
And  of  the  angles  at  the  base,  the  less  is  next  the  greater 
side.  If  BP  is  greater  than  AP,  the  angle  BOP  is  greater 
than  AGP  ;  and  B  is  less  than  A.  (Art.  77.) 


148.  To  enable  us  to  find  the  sides  and  angles  of  an 
oblique  angled  triangle,  three  of  them  must  be  given.  (Art. 
114.) 

These  may  be,  either 

1.  Two  angles  and  a  side,  or 

2.  Two  sides  and  an  angle  opposite  one  of  them,  or 

3.  Two  sides  and  the  included  angle,  or 

4.  The  three  sides. 

The  two  first  of  these  cases  are  solved  by  Theorem  I,  (Art 
143.)  the  third  by  Theorem  II,  (Art.  144.)  and  the  fourth  by 
Theorem  III.  (Art.  145.) 

149  In  making  the  calculations,  it  must  be  kept  in  mind, 
that  the  greater  side  is  always  opposite  to  the  greater  angle, 
(Euc  18,  19.  1.*)  that  there  can  be  only  one  obtuse  angle  hi 
a  triangle,  (Art.  76.)  and  therefore,  that  the  angles  opposite 
to  the  two  least  sides  must  be  acute. 

CASE  I. 

150.  Given, 

Two  angles,  and  )  (  The  remaining  angle,  and 

A  side,  )  i  (  The  other  tw 

*  Thomson's  Legendre,  13.  1. 


OBLIQUE    ANGLED    TRIANGLES. 


The  third  angle  is  found  by  merely  subtracting  the  sum 
of  the  two  which  are  given  from  180°.  (Art.  79.) 

The  sides  are  found,  by  stating,  according  to  Theorem  I, 
the  following  proportion ; 

As  the  sine  of  the  angle  opposite  the  given  side, 
To  the  length  of  the  given  side ; 
So  is  the  sine  of  the  angle  opposite  the  required  side 
To  the  length  of  the  required  side. 

As  a  side  is  to  be  found,  it  is  necessary  to  begin  with  a 
tabular  number. 

Ex.  1.  In  the  triangle  ABC,  the 
side  b  is  given  32  rods,  the  angle  A 
56°  20',  and  the  angle  C  49°  10',  to 
find  the  angle  B,  and  the  sides  a 
and  c. 

The  sum  of  the  two  given  angles 
56°  20'+49°  10'=105°  30' ;  which 
subtracted  from  180°,  leaves  74°  30' 
the  angle  B.  Then, 

Sin  A  !  a 


Calculation  by  logarithms. 


As  the  sine  of  B 
To  the  side  b 
So  is  the  sine  of  A 
To  the  side  a 

As  the  sine  of  B 
To  the  side  b 
So  is  the  sine  of  C 
To  the  side  c 


74°  30;  a.  c. 

0.01609 

32 

1.50515 

56°  20' 

9.92027 

27.64 

1.44151 

74°  30'  a.  c. 

0.01609 

32 

1.50515 

49°  1(V 

9.87887 

25.13 

1.40011 

The  arithmetical  complement  used  in  the  first  term  here, 


02  OBLIQUE    ANGLED    TRIANGLES. 

may  be  found  in  the  usual  way,  or  by  taking  out  the  cose- 
cant of  the  given  angle,  and  rejecting  10  from  the  index. 
(Art.  113.) 

Ex.  2.  Given  the  side  b  71,  the  angle  A  107°  6',  and  the 
angle  C  27°  40'  ;  to  find  the  angle  B,  and  the  sides  a  and  c. 
The  angle  B  is  45°  14'.  Then, 

Sin  A  :  0=95.58 
Sin  B  .  b  . 


When  one  of  the  given  angles  is  obtuse,  as  in  this  exam- 
ple, the  sine  of  its  supplement  is  to  be  taken  from  the  tables. 
(Art.  99.) 

CASE  H. 

151.  Given, 

Two  sides,  and          )  (  The  remaining  side  and 

An  opposite  angle,    )  (  The  other  two  angles. 

One  of  the  required  angles  is  found,  by  beginning  with  a 
side,  and,  according  to  Theorem  I,  stating  the  proportion, 

As  the  side  opposite  the  given  angle, 
To  the  sine  of  that  angle  ; 
So  is  the  side  opposite  the  required  angle, 
To  the  sine  of  that  angle. 

The  third  angle  is  found,  by  subtracting  the  sum  of  the 
other  two  from  180°  ;  and  the  remaining  side  is  found,  by 
the  proportion  in  the  preceding  article. 

152.  In  this  second  case,  if  the  side  opposite  to  the  given 
angle  be  shorter  than  the  other  given  side  the  solution  will 
be  ambiguous.     Two  different  triangles  may  be  formed,  each 
of  which  will  satisfy  the  conditions  of  the  problem. 


OBLIQUE   ANGLED    TRIANGLES. 

Let  the  side  b,  the  angle 
A,  and  the  length  of  the 
side  opposite  this  angle  be 
given.  With  the  latter  for 
radius,  (if  it  be  shorter 
than  6,)describe  an  arc,  cut- 
ting the  line  AH  in  the 

points  B  and  B'.  The  lines  BC  and  B'C,  will  be  equal.  So 
that,  with  the  same  data,  there  may  be  formed  two  different 
triangles,  ABC  and  AB'C. 

There  will  be  the  same  ambiguity  in  the  numerical  calcu- 
lation. The  answer  found  by  the  proportion  will  be  the  sine 
of  an  angle.  But  this  may  be  the  sine  either  of  the  acute 
angle  AB'C,  or  of  the  obtuse  angle  ABC.  For,  BC  being 
equal  to  B'C,  the  angle  CB'B  is  equal  to  CBB'.  Therefore 
ABC,  which  is  the  supplement  of  CBB',  is  also  the  supple- 
ment of  CB'B.  But  the  sine  of  an  angle  is  the  same,  as  the 
sine  of  its  supplement.  (Art.  90.)  The  result  of  the  calcu- 
lation will,  therefore,  be  ambiguous.  In  practice,  however, 
there  will  generally  be  some  circumstances  which  will  deter- 
mine whether  the  angle  required  is  acute  or  obtuse. 

If  the  side  opposite  the  given  angle  be  longer  than  the 
other  given  side,  the  angle  which  is  subtended  by  the  latter, 
will  necessarily  be  acute.  For  there  can  be  but  one  obtuse 
angle  in  a  triangle,  and  this  is  always  subtended  by  the  long- 
est side.  (Art.  149.) 

If  the  given  angle  be  obtuse,  the  other  two  will,  of  course, 
be  acute.  There  can,  therefore,  be  no  ambiguity  in  the 
solution. 

Ex.  1.  Given  the  angle  A,  35°  20',  the  opposite  side  a  50, 
and  the  side  b  70 ;  te  find  the  remaining  side,  and  the  other 
two  angles. 

To  find  the  angle  opposite  to  6,  (Art.  151.) 
a  ;  sin  A  : ;  b  ;  sin  B 


04  OBLIQUE    ANGLED    TRIANGLES. 

The  calculation  here  gives  the  acute  angle  AB'C  54°  3 
50",  and  the  obtuse  angle  ABC  125°  56'  10".  If  the  latter 
be  added  to  the  angle  at  A  35°  20',  the  sum  will  be  161°  16; 
10",  the  supplement  of  which,  18°  43'  50",  is  the  angle 
ACB.  Then  in  the  triangle  ABC,  to  find  the  side  c=AB, 

Sin  A  :  a  :  :  sin  C  ;  c=27.76 

If  the  acute  angle  AB'C  54°  3'  50"  be  added  to  the  angle 
at  A  35°  20',  the  sum  will  be  89°  23'  50",  the  supplement 
of  which,  90°  36'  10",  is  the  angle  ACB'.  Then,  in  the  tri- 
angle AB'C, 

Sin  A  !  CB'  :  :  sin  C  :  AB'=86.45. 


Ex.  2.  Given  the  angle  at  A, 
63°  35',  the  side  b  64,  and  the 
side  a  72  ;  to  find  the  side  c,  and 
the  angles  B  and  C. 


A  «  B 

a  :  sin  A  :  :  b  :  sin  B=52°  457  25" 
Sin  A  :  a  :  :  sin  C  :  c=72.05 

The  sum  of  the  angles  A  and  B,  is  116°  20'  25",  the  sup- 
plement of  which,  63°  39'  35",  is  the  angle  C. 

In  this  example  the  solution  is  not  ambiguous,  because  the 
side  opposite  the  given  angle  is  longer  than  the  other  given 
side. 

Ex.  3.  In  a  triangle  of  which  the  angles  are  A,  B,  and  C, 
and  the  opposite  sides  a,  b,  and  c,  as  before ;  if  the  angle 
A  be  121°  40',  the  opposite  side  a  68  rods,  and  the  side  b  47 
rods ;  whart  are  the  angles  B  and  C,  and  what  is  the  length 
of  the  side  c  ?  Ans.  B  is  36°  2'  4",  C  22°  17'  56",  and  c 
30.3. 

In  this  example  also,  the  solution  is  not  ambiguous,  be- 
cause the  given  angle  is  obtuse. 


OBLIQUE    ANGLED    TRIANGLES.  05 

CASE  III. 
153.  Given, 

Two  sides,  and  )  (  The  remaining  side,  and 

The  included  angle,     )   l  (  The  other  two  angles. 

In  this  case,  the  angles  are  found  by  Theorem  II.  (Art. 
144.)     The  required  side  may  be  found  by  Theorem  I. 

In  making  the  solutions,  it  will  be  necessary  to  observe, 
that  by  subtracting  the  given  angle  from  180°,  the  sum  of 
the  other  two  angles  is  found ;  (Art.  79.)  and,  that  adding 
half  the  difference  of  two  quantities  to  their  half  sum  gives  the 
greater  quantity,  and  subtracting  the  half  difference  from  the 
half  sum  gives  the  less.  The  latter  proposition  may  be  geo- 
metrically demonstrated  thus ; 
Let  AE,  be  the 

greater     of    two 

magnitudes,    and 

BE  the  less.     Bisect  AB  in  D,  and  make  AC  equal  to  BE. 

Then, 

AB  is  the  sum  of  the  two  magnitudes  ; 
CE  their  difference  ; 
DA  or  DB  half  their  sum  ; 
DE  or  DC  half  their  difference  ; 
But  DA-f-DE=AE  the  greater  magnitude  ; 
And  DE— DE=BE  the  less. 
Ex.    1.      In   the    triangle 
ABC,  the  angle  A  is  given  26° 
14',  the  side  b  39,  and   the 
side  c  53 ;  to  find  the  angles  B 
and  C,  and  the  side  a. 

The  sum  of  the  sides  b  and  c  is        63+39=92 
And  their  difference  53 — 39=14 

The  sum  of  the  angles  B  and  C=«1800 — 26°  14'«153°  46' 
And  half  the  sum  of  B  and  C  is  76°  65? 


96  OBLIfcUE    ANGLED    TRIANGLES. 

Then,  by  Theorem  II,  (Fig.  30.) 
(b+c)  :  (b—c)  :  :  tan  i(B+C)  :  tan  £  (B~C) 

To  and  from  the  half  sum  76°  53' 

Adding  and  subtracting  the  half  difference  33     8  50 

We  have  the  greater  angle  110     1  50 

And  the  less  angle  43  44  10 

As  the  greater  of  the  two  given  sides  is  c,  the  greater 
angle  is  C,  and  the  less  angle  B.  (Art.  149.) 

To  find  the  side  a,  by  Theorem  I. 

Sin  B  :  b  :  :  sin  A  :  a=24.94. 

Ex.  2.  Given  the  angle  A  101°  30',  the  side  b  76,  and  the 
side  c  109 ;  to  find  the  angles  B  and  C,  and  the  side  a. 
B  is  30°  57£',  C  47°  32-J',  and  a  144.8 

CASE  IV. 

154.  Given  the  three  sides,  to  find  the  angles. 

In  this  case,  the  solutions  may  be  made,  by  drawing  a  per- 
pendicular to  the  longest  side,  from  the  opposite  angle.  This 
will  divide  the  given  triangle  into  two  ri-ght  angled  triangles. 
The  two  segments  may  be  found  by  Theorem  III.  (Art.  145.) 

There  will  then  be  given,  in  each  of  the  right  angled  tri- 
angles, the  hypothenuse  and  one  of  the  legs,  from  which  the 
angles  may  be  determined,  by  rectangular  trigonometry. 
(Art.  135.) 

Ex.  1.  In  the  triangle  ABC, 
the  side  AB  is  39,  AC  35,  and 
BC  27.  What  are  the  angles  ? 

Let  a  perpendicular  be  drawn 
from  C,  dividing  the  longest  side 
AB  into  the  two  segments  AP 
and  BP.  Then  by  Theorem  III, 

AB  :  AC+BC  :  :  AC— BC  .  AP— BP. 


OBLIQUE    ANGLED    TRIANGLES.  $7 

As  the  longest  side  39  a.  c.  8.40894 

To  the  sum  of  the  two  others  62           1.79239 

So  is  the  difference  of  the  latter  8           0.90309 

To  the  difference  of  the  segments  12.72           1.10442 

The  greater  of  the  two  segments  is  AP,  because  it  is  next 
the  side  AC,  which  is  greater  than  BC.  (Art.  146.) 

To   and   from  half  the  sum  of  the  segments  19.5 

Adding  and  subtracting  half  their  difference,  (Art.  153.)  6.36 
We  have  the  greater  segment  AP  25.86 

And  the  less  BP  13.14 

Then,  in  each  of  the  right  angled  triangles  APC  and  BPC, 
we  have  given  the  hypothenuse  and  base  ;  and  by  Art,  135. 

AC  :  R  :  :  AP  :  cos  A=42°  21'  57" 
BC  :  R  :  :  BP  ;  cos  B=60°  52'  42" 

And  subtracting  the  sum  of  the  angles  A  and  B  from 
180°,  we  have  the  remaining  angle  ACB=76°  45'  21". 

Ex.  2.  If  the  three  sides  of  a  triangle  are  78,  96,  and 
104  ;  what  are  the  angles  ? 

Ans.  45°  41'  48",  61°  43'  27",  and  72°  34'  45". 

Examples  for  Practice. 

1.  Given  the  angle  A  54°  30',  the  angle  B  63°  10',  and  the 
side  a  164  rods;  to  find  the  angle  C,  and  the  sides  6 
and  c. 

4.  Given  the  angle  A  45°  6',  the  opposite  side  a  93,  and  the 
side  b  108  ;  to  find  the  angles  B  and  C,  and  the  side  c. 

3.  Given  the  angle  A  67°  24',  the  opposite  side  a  62,  and 

the  side  b  46  ;  to  find  the  angles  B  and  C,  and  the  side  c. 

4.  Given  the  angle  A  127°  42',  the  opposite  side  a  381,  and 

the  side  b  184 ;  to  find  the  angles  B  and  C,  and  the 
side  c, 

9 


OBLIQUE    ANGLED    TRIANGLES. 


5.  Given  the  side  b  58,  the  side  c  67,  and  the  included  angl* 

A=36° ;  to  find  the  angles  B  and  C,  and  the  side  a. 

6.  Given  the  three  sides,  631,  268,  and   546;  to  find  the 

angles. 

155.  The  three   theorems  demonstrated   in  this  section, 
have  been  here  applied  to  oblique 
angled  triangles  only.     But  they 
are  equally   applicable   to   right 
angled  triangles. 

Thus,  in  the  triangle  ABC,  ac- 
cording to  Theorem  I,  (Art.  143.) 

i> 

Sin  B  :  AC  :  :  sin  A  :  BC 

This  is  the  same  proportion  as  one  stated  in  Art  134,  ex- 
cept that,  in  the  first  term  here,  the  sine  of  B  is  substituted 
for  radius.  But,  as  B  is  a  right  angle,  its  sine  is  equal  to 
radius.  (Art.  95.) 

Again,  in  the  triangle  ABC, 
by  the  same  theorem  ; 

Sin  A  :  BC  :  :  sin  C  :  AB 

This  is  also  one  of  the  pro- 
portions in  rectangular  trigo- 
nometry, when  the  hypothe- 
nuse  is  made  radius. 

The  other  two  theorems  might  be  applied  to  the  solution  of 
right  angled  triangles.  But,  when  one  of  the  angles  is  known 
to  be  a  right  angle,  the  methods  explained  in  the  preceding 
section,  are  much  more  simple  in  practice. 


GEOMETRICAL   CONSTRUCTION    OF    TRIAN&LES.  99 


SECTION  V. 

GEOMETRICAL     CONSTRUCTION     OF     TRIANGLES,     B¥      TUB 
PLANE    SCALE. 

ART.  156.  To  facilitate  the  construction  of  geometrical 
figures,  a  number  of  graduated  lines  are  put  upon  the  com- 
mon two  feet  scale ;  one  side  of  which  is  called  the  Plane 
Scale,  and  the  other  side,  Gunter's  Scale.  The  most  im- 
portant of  these  are  the  scales  of  equal  parts,  and  the  line 
of  chords.  In  forming  a  given  triangle,  or  any  other  right 
lined  figure,  the  parts  which  must  be  made  to  agree  with  the 
conditions  proposed,  are  the  lines  and  the  angles.  For  the 
former,  a  scale  of  equal  parts  is  used ;  for  the  latter,  a  line 
of  chords. 

157.  The  line  on  the  upper  side  of  the  plane  scale,  is 
divided  into  inches  and  tenths  of  an  inch.  Beneath  this,  on 
the  left  hand,  are  two  diagonal  scales  of  equal  parts,*  divided 
into  inches  and  half  inches,  by  perpendicular  lines.  On  the 
larger  scale,  one  of  the  inches  is  divided  into  tenths,  by  lines 
which  pass  obliquely  across,  so  as  to  intersect  the  parallel 
lines  which  run  from  right  to  left.  The  use  of  the  oblique 
lines  is  to  measure  hundredths  of  an  inch,  by  inclining  more 
and  more  to  the  right,  as  they  cross  each  of  the  parallels. 

To  take  off,  for  instance,  an  extent  of  3  inches,  4  tenths, 
and  6  hundredths ; 

Place  one  foot  of  the  dividers  at  the  intersection  of  the 
perpendicular  line  marked  3  with  the  parallel  line  marked  6, 

*  These  lines  are  not  represented  by  a  figure,  as  the  learner  is  rap- 
posed  to  have  the  scale  before  him, 


100  GEOMETRICAL    CONSTRUCTION    OF   TRIANGLES. 

and  the  other  foot  at  the  intersection  of  the  latter  with  the 
oblique  line  marked  4. 

The  other  diagonal  scale  is  of  the  same  nature.  The 
divisions  are  smaller,  and  are  numbered  from  left  to  right. 

158.  In  geometrical  constructions,  what  is  often  required, 
is  to  make  a  figure,  not  equal  to  a  given  one,  but  only  sim- 
ilar.    Now  figures  are  similar  which  have  equal  angles,  and 
the  sides  about  the  equal  angles  proportional.     (Euc.   Def. 
1.  6.*)     Thus  a  land  surveyor,  in  plotting  a  field,  makes  the 
several  lines  in  his  plan  to  have  the  same  proportion  to  each 
other,  as  the  sides  of  the  field.     For  this  purpose  a  scale  of 
equal  parts  may  be  used,  of  any  dimensions  whatever.     If 
the  sides  of  the  field  are  2,  5,  7,  and  10  rods,  and  the  lines 
in  the  plan  are  2,  6,  7,  and  10  inches,  and  if  the  angles  are 
the  same  in  each,  the  figures  are  similar.     One  is  a  copy  of 
the  other,  upon  a  smaller  scale. 

So  any  two  right  lined  figures  are  similar,  if  the  angles  are 
the  same  in  both,  and  if  the  number  of  smaller  parts  in  each 
side  of  one,  is  equal  to  the  number  of  larger  parts  in  the  cor- 
responding sides  of  the  other.  The  several  divisions  on  the 
scale  of  equal  parts  may,  therefore,  be  considered  as  repre- 
senting any  measures  of  length,  as  feet,  rods,  miles,  &c.  All 
that  is  necessary  is,  that  the  scale  be  not  changed,  in  the 
construction  of  the  same  figure ;  and  that  the  several  divi- 
sions and  subdivisions  be  properly  proportioned  to  each 
other.  If  the  larger  divisions,  on  the  diagonal  scale,  are 
units,  the  smaller  ones  are  tenths  and  hundredths.  If  the 
larger  are  tens,  the  smaller  are  units  and  tenths. 

159.  In  laying  down  an  angle,  of  a  given  number  of  de- 
grees, it  is  necessary  to  measure  it.     Now  the  proper  meas- 
ure of  an  angle  is  an  arc  of  a  circle.  (Art.  74.)     And  the 
measure  of  an  arc,  where  the  radius  is  given,  is  its  chord. 
For  the  chord  is  the  distance,  in  a  straight  line,  from  one 

STT"""1  • 

*  Thomson's  Legendre,  Def.  B.  4. 


GEOMETRICAL    CONSTRUCTION    OF   TRIANGLES.  101 

end  of  the  arc  to  the  other. 

Thus  the    chord    AB,    is   a  F\ 

measure   of    the   arc   ADB, 

and  of  the  angle  ACB. 

To  form  the  line  of  chords, 
a  circle  is  described,  and  the 
length  of  its  chords  deter-  ° 

mined  for  every  degree  of  the  quadrant.     These  measures 
are  put  on  the  plane  scale,  on  the  line  marked  CHO. 

160.  The  chord  of  60°  is  equal  to  radius.  (Art.  95.)     In 
laying  down  or  measuring  an  angle,  therefore,  an  arc  must 
be  drawn,  with  a  radius  which  is  equal  to  the  extent  from  0 
to  60  on  the  line  of  chords.     There  are  generally  on  the 
scale,  two  lines  of  chords.     Either  of  these  may  be  used ; 
but  the  angle  must  be  measured  by  the  same  line  from 
which  the  radius  is  taken. 

161.  To  make  an  angle,  then,  of  a  given  number  of  de- 
grees ;  from  one  end  of  a  straight  line  as  a  centre,  and  with 
a  radius  equal  to  the  chord  of  60°  on  the  line  of  chords,  de- 
scribe an  arc  of  a  circle  cutting  a  straight  line.     From  the 
point  of  intersection,  extend  the  chord  of  the  given  number 
of  degrees,  applying  the  other  extremity  to  the  arc ;  and 
through  the  place  of  meeting,  draw  the  other  line  from  the 
angular  point. 

If  the  given  angle  is  obtuse,  take  from  the  scale  the  chord 
of  half  the  number  of  degrees,  and  apply  it  twice  to  the  arc. 
Or  make  use  of  the  chords  of  any  two  arcs  whose  sum  is 
equal  to  the  given  number  of  degrees. 

A  right  angle  may  be  constructed,  by  drawing  a  perpen- 
dicular without  using  the  line  of  chords. 

Ex.  1.  To  make  an  angle  of  32  degrees.  With  the  point 
C,  in  the  line  CH,  for  a  centre,  and  with  the  chord  of  60° 
for  radius,  describe  the  arc  ADF.  Extend  the  chord  of  32° 
from  A  to  B  ;  and  through  B,  draw  the  line  BC.  Then  is 
ACB  an  angle  of  32  degrees. 

9* 


102  GEOMETRICAL    CONSTRUCTION    OF    TRIANGLES. 

2.  To  make  an  angle  of  140  degrees.     On  the  line  CH, 
with  the  chord  of  60°,  describe  the  arc  ADF ;  and  extend 


the  chord  of  70°  from  A  to  D,  and  from  D  to  B.     The  arc 
ADB=70°X2=1400. 

On  the  other  hand  : 

162.  To  measure  an  angle ;  On  the  angular  point  as  a 
centre,  and  with  the  chord  of  60°  for  radius,  describe  an  arc 
to  cut  the  two  lines  which  include  the  angle,     The  distance 
between  the   points  of  intersection,  applied  to  the  line  of 
chords,  will  give  the  measure  of  the  angle  in  degrees.  If  the 
angle  be  obtuse,  divide  the  arc  into  two  parts. 

Ex.  1.  To  measure  the  angle  ACB.  (Fig.  33,  page  101.) 
Describe  the  arc  ADF,  cutting  the  lines  CH  and  CB.  The 
distance  AB,  will  extend  32°  on  the  line  of  chords. 

2.  To  measure  the  angle  ACB.  (Fig.  34.)  Divide  the 
arc  ADB  into  two  parts,  either  equal  or  unequal,  and  meas- 
ure each  part,  by  applying  its  chord  to  the  scale.  The  sum 
of  the  two  will  be  140°. 

163.  Besides  the  lines  of  chords,  and  of  equal  parts,  on 
the  plane  scale ;  there  are  also  lines  of  natural  sines,  tangents, 
and  secants,  marked  Sin.,  Tan.,  and  Sec. ;  of  semitangents, 
marked  S.  T. ;   of   longitude,  marked  Lon.  or   M.  L. ;    of 
rhumbs,  marked  Rhu.  or  Rum.,  &c.     These  are  not  neces- 
sary  in  trigonometrical  construction.    Some  of  them  are  used 
in  Navigation  ;  and  some  of  them  in  the  projections  of  the 
Sphere. 


GEOMETRICAL    CONSTRUCTION    OF   TRIANGLES.  103 

164.  In  Navigation,  the  quadrant,  instead  of  being  grad- 
uated in  the  usual  manner,  is  divided  into  eight  portions, 
called  Rhumbs.     The  Rhumh  lime,  on  the  scale,  is  a  line  of 
chords,  divided  into  rhumbs  and  quarter-rhumbs,  instead  of 
degrees. 

165.  The  line  of  Longitude  is  intended  to  show  the  num- 
ber of  geographical  miles  in  a  degree  of  longitude,  at  differ- 
ent distances  from  the  equator.     It  is  placed   over  the  line 
of  chords,  with  the  numbers  in  an  inverted  order:  so  that 
the  figure  above  shows  the  length  of  a  degree  of  longitude, 
in  any  latitude  denoted  by  the  figure  below.*     Thus,  at  the 
equator,  where  the  latitude  is  0,  a  degree  of  longitude  is  60 
geographical  miles.     In  latitude  40,  it  is  46  miles ;  in  lati- 
tude 60,  30  miles,  &c. 

166.  The  graduation  on  the  line  of  secants  begins  where 
the  line  of  sines  ends.     For  the  greater  sine  is  only  equal  to 
radius  ;    but   the   secant   of  the  least  arc  is  greater  than 
radius. 

167.  The  semitangents  are  the  tangents  of  half  the  given 
arcs.     Thus,  the  semitangent  of  20°  is  the  tangent  of  10°. 
The  line  of  semitangents  is  used  in  one  of  the  projections  of 
the  sphere. 


168.  In  the  construction  of  triangles,  the  sides  and  angles 
which  are  given,  are  laid  down  according  to  the  directions  in 
Arts.  158,  161.     The  parts  required  are  then  .measured,  ac- 
cording to  Arts.  158,  162.     The  following  problems  corres- 
pond with  the  four  cases  of  oblique  angled  triangles ;  (Art. 
148.)  but  are  equally  adapted  to  right  angled  triangles. 

169.  PROB.  I.   The  angles  and  one  side  of  a  triangle  being 
given ;  to  find,  by  construction,  the  other  two  sides. 

Draw  the  given  side.     From  the  ends  of  it,  lay  off  two 


*  Sometimes  the  line  of  longitude  is  placed  under  the  line  of  chords. 


104 


GEOMETRICAL    CONSTRUCTION    OF   TRIANGLES. 


of  the  given  angles.  Extend  the  other  sides  till  they  inter- 
sect; and  then  measure  their  lengths  on  a  scale  of  equal 
parts. 

Ex.  1.  Given  the  side  b  32  rods, 
the  angle  A  56°  20',  and  the  angle 
C  49°  W  ;  to  construct  the  trian- 
gle, and  find  the  lengths  of  the  sides 
a  and  c. 

Their   lengths   will   be   25  and 
27*. 

2.  In   a  right    angled   triangle, 
given  the  hypothenuse  90,  and  the 
angle  A  32°   20',  to  find  the  base 
and  perpendicular. 

The  length  of  AB  will  be  76,  and 
of  BC  48. 

3.  Given  the  side  AC  68,  the  an-    A  B 
gle  A  124°,  and  the  angle  C  37°  : 

to  construct  the  triangle. 

'  170.  PROB.  II.    Two  sides  and  an  opposite  angle  being 
given,  to  find  the  remaining  side,  and  the  other  two  angles. 

Draw  one  of  the  given  sides  ;  from  one  end  of  it,  lay  off 
the  given  angle  ;  and  extend  a  line  indefinitely  for  the  re- 
quired side.  From  the  other  end  of  the  first  side,  with  the 
remaining  given  side  for  radius,  describe  an  are  cutting  the 
indefinite  line.  The  point  of  intersection  will  be  the  end 
of  the  required  side. 

If  the  side  opposite  the  given  angle  be  less  than  the  othei 
given  side,  the  case  will  be  ambig- 
uous. (Art.  152.) 

Ex.  1.  Given  the  angle  A  63° 
35',  the  side  632,  and  the  side  a  36. 

The  side  AB  will  be  36  nearly, 
the  angle  B  52°  45i',  and  C  63° 


GEOMETRICAL   CONSTRUCTION    OF   TRIANGLES.  105 

2.  Given  the.  angle  A 
35°  20',  the  opposite  side 
a  25,  and  the  side  b  35. 

Draw   the    side   b    35, 
make  the  angle  A  35°  20', 

and  extend  AH  indefinite-  _  _ 

ly.     From  C  with  radius 

25,  describe  an  arc  cutting  AH  in  B  and  B'.  Draw  CB  and 
CB',  and  two  triangles  will  be  formed,  ABC  and  AB'C,  each 
corresponding  with  the  conditions  of  the  problem. 

3.  Given  the  angle  A  116°,  the  opposite  side  a  38,  and 
the  side  b  26  ;  to  construct  the  triangle. 

171.  PROS.  III.    Two  sides  and  the  included  angle  being 
given  ;  to  find  the  other  side  and  angles. 

Draw  one  of  the  given  sides.  From  one  end  of  it  lay  off 
the  given  angle,  and  draw  the  other  given  side.  Then  con- 
nect the  extremities  of  this  and  the  first  line. 

Ex.  1.  Given  the  angle  A 
26°  14',  the  side  b  78,  and 
the  side  c  106 ;  to  find  B,  C, 
and  a. 

The  side  a  will  be  50,  the 
angle  B  43°  44',  and  C 
110°  2'. 

2.  Given  A  86°,  b  65,  and  c  83 ;  to  find  B,  C,  and  a. 

172.  PROB.  IV.  The  three  sides  being  given;  to  find  the 
angles. 

Draw  one  of  the  sides,  and  from  one  end  of  it,  with  an 
extent  equal  to  the  second  side,  describe  an  arc.  From  the 
other  end,  with  an  extent  equal  to  the  third  side,  describe  a 
second  arc  cutting  the  first ;  and  from  the  point  of  intersec- 
tion draw  the  two  sides.  (Euc.  22.  1.) 


10'6 


GEOMETRICAL    CONSTRUCTION    OF    TRIAKQL^S. 


Ex.  1.  Given  AB  78,  AC  70, 
and  BC  54,  to  find  the  angles. 

The  angles  will  be  A  42°  22', 
B  60°  52|'  and  C  76°  45i'. 

2.  Given  the  three  sides  58, 
39,  and  46  ;  to  find  the  angles. 

173.  Any  right  lined  figure 

whatever,  whose  sides  and  angles  are  given,  may  be  con- 
structed, by  laying  down  the  sides  from  a  scale  of  equal 
parts,  and  the  angles  from  a  line  of  chords. 

Ex.  Given  the  sides  AB= 
20>  BC=*22,  CD=30,  DE= 
12  ;  and  the  angles  B=102°,    Bf 
C=130°,   D=108°,  to  con- 
struct the  figure. 

frraw   the    side    AB=20, 
make  the  angle  B=102°,  draw 
BC=22,  make  C=130°,   draw  CD=30,  make  D=108°, 
draw  DE=12,  and  connect  E  and  A. 

The  last  line,  EA,  may  be  measured  on  the  scale  of  equal 
parts  j  and  the  angles  E  and.  A,  by  a  line  of  chords. 


GUNTER'S  SCALE.  107 


SECTION   VI. 


IRT.  174,  An  expeditious  method  of  solving  the  problems 
in  /igonometry,  and  making  other  logarithmic  calculations, 
m  <u  mechanical  way,  has  been  contrived  by  Mr.  Edmund 
irunter.  The  logarithms  of  numbers,  of  sines,  tangents, 
<fec.,  are  represented  by  lines.  By  means  of  these,  multipli- 
cation, division,  the  rule  of  three,  involution,  evolution,  &c., 
may  be  p^rfoimed  much  more  rapidly,  than  in  the  usual 
method  by  figures. 

The  logaritharic  lines  are  generally  placed  on  one  side 
only  of  the  scale  in  common  use.  They  are, 

A  line  of  artificial  Sines  divided  into  ^Rhumbs,  and 

marked,  S.  R. 

A  line  of  artificial  Tangent*,  do                   T .  R. 

A  line  of  the  logarithms  of  Numbers,  Num. 

A  line  of  artificial  Sines,  to  every  degree,  SIN. 

A  line  of  artificial  Tangents,  do                 TAN. 

A  line  of    Versed  Sims.  V.  S. 

To  these  are  added  a  line  of  equal  parts,  and  a  line  of 
Meridional  Parts,  which  are  not  logarithmic.  The  latter  is 
used  in  Navigation. 

The  Line  of  Numbers. 

175.  Portions  of  the  line  of  Numbers,  are  intended  to 
represent  the  logarithms  of  the  natural  series  of  numbers 
2,  3,  4,  5,  <fec. 

The  logarithms  of  10,  100,  1000,  &c.,  are  1,  2,  3,  <kc. 
(Art.  3.) 


108  CfI7NTERrS    SCALE, 

If,  then,  the  log.  of  10  be  represented  by  a  line  of  1  foot ; 
the  log.  of  100  will  be  repres'd  by  one  of  2  feet; 
the  log.  of  1000  by  one  of  3  feet ; 

the  lengths  of  the  several  lines  being  proportional  to  the  cor- 
responding logarithms  in  the  tables.  Portions  of  a  foot  will 
represent  the  logarithms  of  numbers  between  1  and  10  ;  and 
portions  of  a  line  2  feet  long,  the  logarithms  of  numbers  be- 
tween 1  and  100. 

On  Gunter's  scale,  the  line  of  the  logarithms  of  numbers 
begins  at  a  brass  pin  on  the  left,  and  the  divisions  are  num- 
bered 1,  2,  3,  <fec.,  to  another  pin  near  the  middle.  From 
this  the  numbers  are  repeated,  2,  3,  4,  &c.,  which  may  be 
read  20,  30,  40,  &c.  The  logarithms  of  numbers  between 
1  and  10,  are  represented  by  portions  of  the  first  half  of  the 
line;  and  the  logarithms  of  numbers  between  10  and  100, 
by  portions  greater  than  half  the  line,  and  less  than  the 
whole. 

176.  The  logarithm  of  1,  which  is  0,  is  denoted,  not  by 
any  extent  of  line,  but  by  a  point  under  1,  at  the  commence- 
ment of  the  scale.  The  distances  from  this  point  to  differ- 
ent parts  of  the  line,  represent  other  logarithms,  of  which 
the  figures  placed  over  the  several  divisions  are  the  natural 
numbers.  For  the  intervening  logarithms,  the  intervals  be- 
tween the  figures,  are  divided  into  tenths,  and  sometimes 
into  smaller  portions.  On  the  right  hand  half  of  the  scale, 
as  the  divisions  which  are  numbered  are  tens,  the  subdivisions 
are  units. 

Ex.  1.  To  take  from  the  scale  the  logarithm  of  3.6;  set 
one  foot  of  the  dividers  under  1  at  the  beginning  of  the 
scale,  and  extend  the  other  to  the  6th  division  after  the  first 
figure  3. 

2.  For  the  logarithm  of  47 ;  extend  from  1  at  the  be- 
ginning, to  the  7th  subdivision  after  the  second  figure  4*. 

*  If  the  dividers  will  not  reach  the  distance  required ;  first  open  them  so 
as  to  take  off  half,  or  any  part  of  the  distance,  and  then  the  remaining  part. 


GtNTER's    SCALE.  109 

177.  It   will   be   observed,  that   the  divisions  and  sub- 
divisions decrease,  from  left  to  right ;  as  in  the  tables  of  logo,' 
rithms,  the  differences  decrease.    The  difference  between  the 
logarithms  of  10  and  100,  is  no  greater,  than  the  difference 
between  the  logarithms  of  1  and  10. 

178.  The  line  of  numbers,  as  it  has  been  here  explained, 
furnishes  the  logarithms  of  all  numbers  between  1  and  100. 

And  if  the  indices  of  the  logarithms  be  neglected,  the 
same  scale  may  answer  for  all  numbers  whatever.  For  the 
decimal  part  of  the  logarithm  of  any  number  is  the  same, 
as  that  of  the  number  multiplied  or  divided  by  10,  100, 
&c.  (Art.  14.)  In  logarithmic  calculations,  the  use  of  the 
indices  is  to  determine  the  distance  of  the  several  figures 
of  the  natural  numbers  from  the  place  of  units.  (Art.  11.) 
But  in  those  cases  in  which  the  logarithmic  line  is  com- 
monly used,  it  will  not  generally  be  difficult  to  determine  the 
local  value  of  the  figures  in  the  result. 

179.  We  may,  therefore,  consider  the  point  under  1  at  the 
left  hand,  as  representing  the  logarithm  of  1,  or  10,  or  100; 
or  iV,  or  T-tar,  &c.,  for  the  decimal  part  of  the  logarithm  of 
each  of  these  is  0.     But  if  the  first  1  is  reckoned  10,  all  the 
succeeding  numbers  must  also  be   increased    in  a  tenfold 
ratio  ;  so  as  to  read,  on  the  first  half  of  the  line,  20,  30,  40, 
&c.,  and  on  the  other  half,  200,  300,  &c. 

The  whole  extent  of  the  logarithmic  line, 

is  from  1      to  100,  or  from  0.1      to  10, 

or  from  10    to  1000,  or  from  0.01    to  1, 

or  from  100  to  10000,  &c.      or  from  0.001  to  0.1,  &c. 

Different  values  may,  on  different  occasions,  be  assigned 
to  the  several  numbers  and  subdivisions  marked  on  this  line. 
But  for  any  one  calculation,  the  value  must  remain  the 
same. 

Ex.  Take  from  the  scale  365.   • 

As  this  number  is  between  10  and  1000,  let  the  1  at  the 
10 


beginning  of  the  scale,  be  reckoned  10.  Then,  from  this 
point  to  the  second  3  is  300 ;  to  the  6th  dividing  stroke  is 
60 ;  and  half  way  from  this  to  the  next  stroke  is  5. 

180.  Multiplication,  division,   &c.,  are  performed  by  the 
line  of  numbers,  on  the  same  principle,  as  by  common  loga- 
rithms.    Thus, 

To  multiply  by  this  line,  add  the  logarithms  of  the  two 
factors;  (Art.  37.)  that  is,  take  off,  with  the  dividers,  that 
length  of  line  which  represents  the  logarithm  of  one  of  the 
factors,  and  apply  this  so  as  to  extend  forward  from  the  end 
of  that  which  represents  the  logarithm  of  the  other  factor. 
The  sum  of  the  two  will  reach  to  the  end  of  the  line  rep- 
resenting the  logarithm  of  the  product. 

Ex.  Multiply  9  into  8.  The  extent  from  1  to  8,  added  to 
that  from  1  to  9,  will  be  equal  to  the  extent  from  1  to  72,  the 
product. 

181.  To  divide  by  the  logarithmic  line,  subtract  the  loga- 
rithm of  the  divisor  from  that  of  the  dividend  ;  (Art.  41.) 
that  is,  take  off  the  logarithm  of  the  divisor,  and  this  extent 
set  back  from  the  end  of  the  logarithm  of  the  dividend,  will 
reach  to  the  logarithm  of  the  quotient! 

Ex.  Divide  42  by  7.  The  extent  from  1  to  7,  set  back 
from  42,  will  reach  to  6,  the  quotient. 

182.  Involution  is  performed  in  logarithms,  by  multiply- 
ing the  logarithm    of    the  quantity  into  the  index  of   the 
power ;  (Art.  45.)  that  is,  by  repeating  the  logarithms  as 
many  times  as  there  are  units  in  the  index.     To  involve  a 
quantity  on  the  scale,  then,  take  in  the  dividers  the  linear 
logarithm,  and  double  it,  treble  it,  &c.,  according  to  the  index 
of  the  proposed  power. 

Ex.  1.  Required  the  square  of  9.  Extend  the  dividers 
from  1  to  9.  Twice  this  extent  will  reach  to  81,  the  square. 

2.  Required  the  cube  of  4.  The  extent  from  1  to  4  re- 
peated three  times,  will  reach  to  64  the  cube  of  4. 

183.  On  the  other  hand,  to  perform  evolution  on  the  scale  ; 


111 

take  half,  one-third,  <kc.,  of  the  logarithm  of  the  quantity,  ac- 
cording to  the  index  of  the  proposed  root. 

Ex.  1.  Required  the  square  root  of  49.  Half  the  extent 
from  1  to  49,  will  reach  from  1  to  7,  the  root. 

2.  Required  the  cube  root  of  27.  One  third  the  distance 
from  1  to  27,  will  extend  from  1  to  3,  the  root. 

184.  The  Rule  of  Three  may  be  performed  on  the  scale, 
in  the  same  manner  as  in  logarithms,  by  adding  the  two 
middle  terms,  and  from  the  sum,  subtracting  the  first  term 
(Art.  52.)     But  it  is  more  convenient  in  practice  to  begin  by 
subtracting  the  first  term  from  one  of  the  others.     If  four 
quantities  are  proportional,  the  quotient  of  the  first  divided 
by  the  second,  is  equal  to  the  quotient  of  the  third  divided 
by  the  fourth.  (Alg.  315.) 

Thus,  if  a  ;  6  :  :  c  :  d,  theni=-,  and^LJ-.  (Alg.  344.) 
6     d         c     d 

But  in  logarithms,  subtraction  takes  the  place  of  division ; 
so  that, 

log.  a — log.  6=log.  c — log.  d.    Or,  log.  a — log.  c=log.  b — 
log.  d. 

Hence, 

185.  On   the  scale,  the  difference   between    the  first  and 
second  terms  of  a  proportion,  is  equal  to  the  difference  between 
the  third  and  fourth.     Or,  the  difference  between  the  first 
and   third  terms,  is  equal   to  the  difference  between   the 
second  and  fourth. 

The  difference  between  the  two  terms  is  taken,  by  ex- 
tending the  dividers  from  one  to  the  other.  If  the 
second  term  be  greater  than  the  first ;  the  fourth  must  be 
greater  than  the  third;  if  less,  less.*  Therefore,  if  the 
dividers  extend  forward  from  left  to  right,  that  is,  from  a 
less  number  to  a  greater,  from  the  first  term  to  the  second ; 

*  Euclid,  14,5. 


112  GUNTER'S  SCALE. 

they  must  also  extend  forward  from  the  third  to  the  fourth. 
But  if  they  extend  backward,  from  the  first  term  to  the 
second  ;  they  must  extend  the  same  way,  from  the  third  to 
the  fourth. 

Ex.  1.  In  the  proportion  3  :  8  t  :  12  :  32,  the  extent 
from  3  to  8,  will  reach  from  12  to  32 ;  Or,  the  extent  from 
8  to  12,  will  reach  from  8  to  32. 

2.  If  54  yards  of  cloth  cost  48  dollars,  what  will  18  yds. 
cost? 

54  :  48  :  :  18  :  16 

The  extent  from  64  to  48,  will  reach  backwards  from  18 
to  16, 

3.  If  63  gallons  of  wine  cost  81  dollars,  what  will  35  gal- 
lons cost  ? 

63  :  81  :  :  35  :  45 
The  extent  from  63  to  81,  will  reach  from  35  to  45. 

The  Line  of  Sines. 

186.  The  line  on  Gunter's  scale  marked  SIN.  is  a  line  of 
logarithmic  sines,  made  to  correspond  with  the  line  of  num- 
bers. The  whole  extent  of  the  line  of  numbers,  (Art,  179.) 

is  from  1  to  100,  whose  logs,  are  0.00000  and  2.00000, 
or  from  10  to  1000,  whose  logs,  are  1.00000  and  3.00000, 
or  from  100  to  10000,  whose  logs,  are  2.00000  4.00000, 

the  difference  of  the  indices  of  the  two  extreme  logarithms 
being  in  each  case  2. 

Now  the  logarithmic  sine  of  0°  34'  22"  41"'  is    8.00000 
And  the  sine  of  90°  (Art.  95.)  is  10.00000 

Here  also  the  difference  of  the  indices  is  2.  If  then  the 
point  directly  beneath  one  extremity  of  the  line  of  numbers, 
be  marked  for  the  sine  of  0°  34'  22"  41"' ;  and  the  point 


GUNTER'S  SCALE.  113 

beneath  the  other  extremity,  for  the  sine  of  90°  ;  the  interval 
may  furnish  the  intermediate  sine ;  the  divisions  on  it  being 
made  to  correspond  with  the  decimal  part  of  the  logarithmic 
sines  in  the  tables.* 

The  first  dividing  stroke  in  the  line  of  Sines  is  generally 
at  0°  40',  a  little  farther  to  the  right  than  the  beginning  of 
the  line  of  numbers.  The  next  division  is  at  0°  50' ;  then 
begins  the  numbering  of  the  degrees,  1,  2,  3,  4,  &c.,  from 
left  to  right. 

The  Line  of  Tangents. 

187.  The  first  45  degrees  on  this  line  are  numbered  from 
left  to  right,  nearly  in  the  same  manner  as  on  the  line  of 
Sines. 

The  logarithmic  tangent  of  0°  34'  22"  35'"  is    8.00000 
And  the  tangent  of  45°,  (Art.  95.)  is  10.00000 

The  difference  of  the  indices  being  2,  45  degrees  will 
reach  to  the  end  of  the  line.  For  those  above  45°  the  scale 
ought  to  be  continued  much  farther  to  the  right.  But  as 
this  would  be  inconvenient,  the  numbering  of  the  degrees, 
after  reaching  45,  is  carried  back  from  right  to  left.  The 
same  dividing  stroke  answers  for  an  arc  and  its  complement, 
one  above  and  the  other  below  45°.  For,  (Art.  93.  Pro- 
por.  9.) 

tan  :  R  :  :  R  :  cot 
In  logarithms,  therefore,  (Art.  184.) 
tan  — R=R  — cot. 

That  is,  the  difference  between  the  tangent  and  radius,  is 
equal  to  the  difference  between  radius  and  the  cotangent :  in 

*  To  represent  the  sines  less  than  34'  22"  41'",  the  scale  must  be  ex- 
tended on  the  left  indefinitely.  For,  as  the  sine  of  an  arc  approaches 
to  0,  its  logarithm,  which  is  negative,  increases  without  limit.  (Art.  15.) 


114 

other  words,  one  is  as  much  greater  than  the  tangent  of  45°, 
as  the  other  is  less.  In  taking,  then,  the  tangent  of  an  arc 
greater  than  45°,  we  are  to  suppose  the  distance  between 
45  and  the  division  marked  with  a  given  number  of  degrees, 
to  be  added  to  the  whole  line,  in  the  same  manner  as  if  the 
line  were  continued  out.  In  working  proportions,  extending 
the  dividers  lack,  from  a  less  number  to  a  greater,  must 
be  considered  the  same  as  carrying  them  forward  in  other 
cases.  See  Art.  185. 

Trigonometrical  Proportions  on  the  Scale. 

188.  In  working  proportions  in  trigonometry  by  the  scale; 
the  extent  from  the  first  term  to  the  middle  term  of  the  same 
name,  will  reach  from  the  other  middle  term  to  the  fourth 
term.  (Art.  185.) 

In  a  trigonometrical  proportion,  two  of  the  terms  are  the 
lengths  of  sides  of  the  given  triangle  ;  and  the  other  two  are 
tabular  sines,  tangents,  &c.  The  former  are  to  be  taken  from 
the  line  of  numbers ;  the  latter,  from  the  lines  of  logarith- 
mic sines  and  tangents.  If  one  of  the  terms  is  a  secant,  the 
calculation  cannot  be  made  on  the  scale,  which  has  com- 
monly no  line  of  secants.  It  must  be  kept  in  mind  that 
radius  is  equal  to  the  sine  of  90°,  or  to  the  tangent  of  45°. 
(Art.  95.)  Therefore,  whenever  radius  is  a  term  in  the  pro- 
portion, one  foot  of  the  dividers  must  be  set  on  the  end  of 
the  line  of  sines  or  of  tangents. 

189.  The  following  examples   are  taken  from  the  propor- 
tions which  have  already  been  solved  by  numerical  calcula- 
tion. 

Ex.  1.  In  Case  I,  of  right  angled  triangles,  (Art.  134. 
ex.  1.) 

R  I  45  :  :  sin  32°  20'  :  24 

Here  the  third  term  is  a  sine  ;  the  first  term  radius  is, 
therefore,  to  be  considered  as  the  sine  of  90°.  Then  the 


GUNTER'S  SCALE.  115 

extent  from  90°  to  32°  20'  on  the  line  of  sines,  will  reach 
from  45  to  24  on  the  line  of  numbers.  As  the  dividers 
are  set  back  from  90°  to  32°  20' ;  they  must  also  be  set  back 
from  45.  (Art.  185.) 

2.  In  the  same  case,  if  the  base  be  made  radius,  (page  60.) 

R  :  38  :  :  tan  32°  20'  ;  24 

Here,  as  the  third  term  is  a  tangent,  the  first  term  radius 
is  to  be  considered  the  tangent  of  45°.  Then  the  extent 
from  45°  to  32°  20'  on  the  line  of  tangents,  will  reach  from 
38  to  24  on  the  line  of  numbers. 

3.  If  the  perpendicular  be  made  radius,  (page  62.) 

R  t  24  :  :  tan  57°  40;  :  38 

The  extent  from  45°  to  57°  40'  on  the  line  of  tangents, 
will  reach  from  24  to  38  on  the  line  of  numbers.  For  the 
tangent  of  5*7°  40'  on  the  scale,  look  for  its  complement  32° 
20'.  (Art.  187.)  In  this  example,  although  the  dividers 
extend  back  from  45°  to  57°  40' ;  yet,  as  this  is  from  a  less 
number  to  a  greater,  they  must  extend  for  ward  on  the  line 
of  numbers.  (Arts.  185,  187.) 

4.  In  Art.  135, 

35  :  R  :  :  26  :  sin  48° 
The  extent  from  35  to  26  will  reach  from  90°  to  48°. 

5.  In  Art.  136, 

R  I  48  :  :  tan  27-J-0  :  24| 
The  extent  from  45°  to  2^|°,  will  reach  from  48  to  24f. 

6.  In  Art.  150,  ex.  1. 

Sin  74°  30'  :  32  :  :  sin  56°  20'  :  27£. 

For  other  examples,  see  the  several  cases  in  Sections  IEL 
and  IV. 

190.  Though  the  solutions  in  trigonometry  may  be  ef- 


116  TRIGONOMETRICAL   ANALYSIS. 

fected  by  the  logarithmic  scale,  or  by  geometrical  construc- 
tion, as  well  as  by  arithmetical  computation  ;  yet  the  latter 
method  is  by  far  the  most  accurate.  The  first  is  valuable 
principally  for  the  expedition  with  which  the  calculations  are 
made  by  it.  The  second  is  of  use,  in  presenting  the/orm 
of  the  triangle  to  the  eye.  But  the  accuracy  which  attends 
arithmetical  operations,  is  not  to  be  expected,  in  taking  lines 
from  a  scale  with  a  pair  of  dividers.* 


SECTION  VII. 

THE    FIRST    PRINCIPLES    OF    TRIGONOMETRICAL    ANALYSIS. 

ART.  191.  In  the  preceding  sections,  sines,  tangents,  and 
secants  have  been  employed  in  calculating  the  sides  and  an- 
gles of  triangles.  But  the  use  of  these  lines  is  not  confined 
to  this  object.  Important  assistance  is  derived  from  them, 
in  conducting  many  of  the  investigations  in  the  higher 
branches  of  analysis,  particularly  in  physical  astronomy.  It 
does  not  belong  to  an  elementary  treatise  of  trigonometry, 
to  prosecute  these  inquiries  to  any  considerable  extent.  But 
this  is  the  proper  place  for  preparing  the  formula,  the  appli- 
cations of  which  are  to  be  made  elsewhere. 

Positive  and  Negative  SIGNS  in  Trigonometry. 

192.  Before  entering  on  a  particular  consideration  of  the 
algebraic  expressions  which  are  produced  by  combinations 
of  the  several  trigonometrical  lines,  it  will  be  necessary  to 
attend  to  the  positive  and  negative  signs  in  the  different 

*  See  note  0. 


TRIGONOMETRICAL   ANALYSIS. 


quarters  of  the  circle.  The  sines,  tangents,  <kc.,  in  the 
tables,  are  calculated  for  a  single  quadrant  only.  But  these 
are  made  to  answer  for  the  whole  circle.  For  they  are  of  the 
same  length  in  each  of  the  four  quadrants.  (Art.  90.)  Some 
of  them  however,  are  positive;  while  others  are  negative. 
In  algebraic  processes,  this  distinction  must  not  be  neg- 
lected. 

193.  For  the  purpose  of  tracing  the  changes  of  the  signs, 
in  different   parts  of  the  circle,  let  it  be  supposed   that  a 
straight  line    CT   is 
fixed  at  one  end  C, 
while  the  other  end 
is  carried  round,  like 
a  rod  moving  on  a 
pivot ;    so  that   the 
point  S  shall  describe 


the  circle  ABDH.  If 
the  two  diameters 
AD  and  BH,  be  per- 
pendicular to  each 
other,  they  will  di- 
vide the  circle  into 
quadrants. 

194.  In  the  first  quadrant  AB,  the  sine,  cosine,  tangent, 
&c.,  are  considered  all  positive.  In  the  second  quadrant  BD, 
the  sine  P'S'  continues  positive ;  because  it  is  still  on  the 
upper  side  of  the  diameter  AD,  from  which  it  is  measured. 
But  the  cosine,  which  is  measured  from  BH,  becomes  nega- 
tive, as  soon  as  it  changes  from  the  right  to  the  left  of  this 
line.  (Alg.  382.)  In  the  third  quadrant  the  sine  becomes 
negative,  by  changing  from  the  upper  side  to  the  under  side 
of  DA.  The  cosine  continues  negative,  being  still  on  the 
left  of  BH.  In  the  fourth  quadrant,  the  sine  continues  neg- 
ative. But  the  cosine  becomes  positive,  by  passing  to  the 
right  of  BH. 


118  TRIGONOMETRICAL    ANALYSIS. 

195.  The  signs  of  the   tangents  and  secants  may  be  de- 
rived from  those  of  the  sines  and  cosines.     The  relations  of 
these  several  lines  to  each  other  must  be  such,  that  a  uniform 
method  of   calculation  may  extend  through   the  different 
quadrants. 

In  the  first  quadrant,  (Art.  93.  Propor.  1.) 

K  :  cos  t  :  tan  ;  sin,  that  is,  Tan— Rxsin. 

cos 

The  sign  of  the  quotient  is  determined  from  the  signs  of 
the  divisor  and  dividend.  (Alg.  100.)  The  radius  is  con- 
sidered as  always  positive.  If  then  the  sine  and  cosine  be 
both  positive  or  both  negative,  the  tangent  will  be  positive, 
But  if  one  of  these  be  positive,  while  the  other  is  negative, 
the  tangent  will  be  negative. 

Now  by  the  preceding  article, 

In  the  2d  quadrant,  the  sine  is  positive,  and  the  cosine 
negative. 

The  tangent  must  therefore  be  negative. 

In  the  3d  quadrant,  the  sine  and  cosine  are  both  negative. 
The  tangent  must  therefore  be  positive. 

In  the  4th  quadrant,  the  sine  is  negative,  and  the  cosine 
positive. 

The  tangent  must  therefore  be  negative. 

196.  By  the  9th,  3d,  and  6th  proportions  in  Art.  93. 

Ra 

1.  Tan  :  R  :  :  R  :  cot,  that  is  Cot= — • 

tan 

Therefore,  as  radius  is  uniformly  positive,  the  cotangent 
must  have  the  same  sign  as  the  tangent. 

Ra 

2.  Cos  :  R  :  :  R  :  sec,"  that  is,  Sec*- — • 

cos 


TRIGONOMETRICAL    ANALYSIS. 


119 


The  secant,  therefore,  must  have  the  same  sign  as  the 
cosine. 


3.    Sin  :  B  :  :  R  :  cosec,  that  is, 


sn 


The  cosecant,  therefore,  must  have  the  same  sign  as  the 
sine. 

The  versed  sine,  as  it  is  measured  from  A,  in  one  direction 
only,  is  invariably  positive. 

197.  The  tangent 
AT  increases,  as  the  T 
arc  extends  from  A 
towards  B.  (See  also 
Fig  11.  p.  69.)  Near 
B  the  increase  is  very 
rapid  ;  and  when  the 
difference  between 
the  arc  and  90°,  is 
less  than  any  assign- 
able quantity,  the 
tangent  is  greater 
than  any  assignable 
quantity,  and  is  said 

to  be  infinite.  (Alg.  447.)  If  the  arc  is  exactly  90  degrees, 
it  has,  strictly  speaking,  no  tangent.  For  a  tangent  is  a  line 
drawn  perpendicular  to  the  diameter  which  passes  through 
one  end  of  the  arc,  and  extended  till  it  meets  a  line  proceed- 
ing from  the  centre  through  the  other  end.  (Art.  84.)  But 
if  the  arc  is  90  degrees,  as  AB,  the  angle  ACB  is  a  right 
angle,  and  therefore  AT  is  parallel  to  CB  ;  so  that,  if  these 
lines  be  extended  ever  so  far,  they  never  can  meet.  Still,  as 
an  arc  infinitely  near  to  90°  has  a  tangent  infinitely  great,  it 
is  frequently  said,  in  concise  terms,  that  the  tangent  of  90° 
is  infinite. 

In  the  second  quadrant,  the  tangent  is,  at  first,  infinitely 


120 


TRIGONOMETRICAL   ANALYSIS. 


great,  and  gradually 
diminishes,  till  at  D 
it  is  reduced  to  no- 
thing. In  the  third 
quadrant,  it  increases 
again,  becomes  infi- 
nite near  H,  and  is 
reduced  to  nothing 
at  A. 

The  cotangent  is  in- 
versely as  the  tan- 
gent. It  is  there- 
fore nothing  at  B 
and  H,  and  infinite 
near  A  and  D. 

198.  The  secant  increases  with  the  tangent,  through  the 
first  quadrant,  and  becomes  infinite  near  B  ;  it  then  dimin- 
ishes, in  the  second  quadrant,  till  at  D   it  is  equal  to  the 
radius  CD.     In  the  third  quadrant  it  increases  again,  becomes 
infinite  near  H,  after  which  it  diminishes,   till  it  becomes 
equal  to  radius. 

The  cosecant  decreases,  as  the  secant  increases,  and  v.  v. 
It  is  therefore  equal  to  radius  at  B  and  H,  and  infinite  near 
A  and  D. 

199.  The  sine  increases  through  the  first  quadrant,  till  at 
B  it  is  equal  to  radius.  (See  also  Fig.  13.  page  70.)  It  then 
diminishes,  and  is  reduced  to  nothing  at  D.     In  the  third 
quadrant,  it  increases  again,  becomes  equal  to  radius  at  H, 
and  is  reduced  to  nothing  at  A. 

The  cosine  decreases  through  the  first  quadrant,  and  is  re- 
duced to  nothing  at  B.  In  the  second  quadrant,  it  increases, 
till  it  becomes  equal  to  radius  at  D.  It  then  diminishes 
again,  is  reduced  to  nothing  at  H,  and  afterwards  increases 
till  it  becomes  equal  to  radius  at  A. 


TRIGONOMETRICAL   ANALYSIS.  121 

In  all  these  cases,  the  arc  is  supposed  to  begin  at  A,  and 
to  extend  round  in  the  direction  of  BDH, 

200.  The  sine  and  cosine  vary  from  nothing  to  radius, 
which  they  never  exceed.     The  secant  and  cosecant  are  never 
less  than  radius,  but  may  be  greater  than  any  given  length. 

The  tangent  and  cotangent  have  every  value  from  nothing 
to  infinity.  Each  of  these  lines,  after  reaching  its  greatest 
limit,  begins  to  decrease  ;  and  as  soon  as  it  arrives  at  its  least 
limit,  begins  to  increase.  Thus,  the  sine  begins  to  decrease, 
after  becoming  equal  to  radius,  which  is  its  greatest  limit. 
But  the  secant  begins  to  increase  after  becoming  equal  to 
radius,  which  is  its  least  limit. 

201.  The  substance  of  several  of  the  preceding  articles  is 
comprised  in  the  following  tables.    The  first  shows  the  signs 
of  the  trigonometrical  lines,  in  each  of  the  quadrants  of  the 
circle.     The  other  gives  the  values  of  these  lines,  at  the  ex- 
tremity of  each  quadrant. 

Quadrant  1st  2d  3d  4th 

Sine  and  cosecant  -h  +  —  — 

Cosine  and  secant  +  —  —  + 

Tangent  and  cotangent  +  —  +  — 

0°  90°  180°  270°  360° 

Sine  0  r  0  r  0 

Cosine  r  0  r  0  r 

Tangent  0  oc  0  oc  0 

Cotangent  oc  0  oc  0  oc 

Secant  r  oc  r  cc  r 

Cosecant  oc  r  oc  r  oc 

Here  r  is  put  for  radius,  and  oc  for  infinite, 

202.  By  comparing  these  two  tables,  it  will  be  seen,  that 
each  of  the  trigonometrical  lines  changes  from  positive  to 
negative,  or  from  negative  to  positive,  in  that  part  of  the 
circle  in  which  the  line  is  either  nothing  or  infinite.    Thus, 

11 


122  TRIGONOMETRICAL   ANALYSIS. 

the  tangent  changes  from  positive  to  negative,  in  passing 
from  the  first  quadrant  to  the  second,  through  the  place 
where  it  is  infinite.  It  becomes  positive  again,  in  passing 
from  the  second  quadrant  to  the  third,  through  the  point  in 
which  it  is  nothing. 

203.  There  can  be  no  more  than  360  degrees  in  any  circle. 
But  a  body  may  have  a  number  of  successive  revolutions  in 
the  same  circle  ;  as  the  earth  moves  round  the  sun,  nearly  in 
the  same  orbit,  year  after  year.  In  astronomical  calculations, 
it  is  frequently  necessary  to  add  together  parts  of  different 
revolutions.  The  sum  may  be  more  than  360°.  But  a  body 
which  has  made  more  than  a  complete  revolution  in  a  circle, 
is  only  brought  back  to  a  point  which  it  had  passed  over 
before.  So  the  sine,  tangent,  &c.,  of  an  arc  greater  than 
360°,  is  the  same  as  the  sine,  tangent,  &c.,  of  some  arc  less 
than  360°.  If  an  entire  circumference,  or  a  number  of  cir- 
cumferences, be  added  to  any  arc,  it  will  terminate  in  the 
same  point  as  before.  So  that,  if  0  be  put  for  a  whole  cir- 
cumference, or  360°,  and  x  be  any  arc  whatever  ; 

sin  z=sin  (C+z)=sin  (2C+z)=sin  (3  C+x),  &c. 
tan  z=tan  (C+x)==tan  (2  C-f#>=tan 


204.  It  is  evident  also,  that,  in  a  number  of  successive 
revolutions,  in  the  same  circle  ; 

The  first  quadrant  must  coincide  with  the  5th,  9th,  13th,  lYth, 
The  second,  with  the  6th,  10th,  Uth,  18th,  &c. 

The  third,  with  the  7th,  Ilth,  15th,  19th,  <fec. 

The  fourth,  with  the  8th,  12th,  16th,  20th,  &c. 

205.  If  an  arc,  extending  in  a  certain  direction  from  a 
given  point,  be  considered  positive  ;  an  arc  extending  from 
the  same  point,  in  an  opposite  direction,  is  to  be  considered 


TRIGONOMETRICAL   ANALYSIS. 


128 


negative.  (Alg.  382.) 
Thus,  if  the  arc  ex- 
tending from  A  to  S, 
be  positive;  an  arc 
extending  from  A  to 
S;//  will  be  negative. 
The  latter  will  not 
terminate  in  the  same 
quadrant  as  the  other 
— and  the  signs  of 
the  tabular  lines 
must  be  accommo- 
dated to  this  circum- 
stance. Thus,  the 

sine  of  AS  will  be  positive,  while  that  of  AS'"  will  be  nega- 
tive. (Art.  194.)  When  a  greater  arc  is  subtracted  from  a 
less,  if  the  latter  be  positive,  the  remainder  must  be  nega- 
tive. (Alg.  40.) 


TRIGONOMETRICAL    FORMULAE. 

206.  From  the  view  which  has  been  here  taken  of  the 
changes  in  the  trigonometrical  lines,  it  will  be  easy  to  see, 
in  what  parts  of  the  circle  each  of  them  increases  or  de- 
creases. But  this  does  not  determine  their  exact  values,  ex- 
cept at  the  extremities  of  the  several  quadrants.  In  the 
analytical  investigations  which  are  carried  on  by  means  of 
these  lines,  it  is  necessary  to  calculate  the  changes  produced 
in  them,  by  a  given  increase  or  diminution  of  the  arcs  to 
which  they  belong.  In  this  there  would  be  no  difficulty,  if 
the  sines,  tangents,  <fec.,  were  proportioned  to  their  arcs.  But 
this  is  far  from  being  the  case.  If  an  arc  is  doubled,  its 
sine  is  not  exactly  doubled.  Neither  is  its  tangent  or  se- 
cant. We  have  to  inquire,  then,  in  what  manner  the  sine, 


124 


TRIGONOMETRICAL    ANALYSIS. 


tangent,  &c.,  of  one  arc  may  be  obtained,  from  those  of  other 
arcs  already  known. 

The  problem  on  which  almost  the  whole  of  this  branch 
of  analysis  depends,  consists  in  deriving,  from  the  sines  and 
cosines  of  two  given  arcs,  expressions  for  the  sine  and  cosine 
of  their  sum  and  difference.  For,  by  addition  and  subtrac- 
tion, a  few  arcs  may  be  so  combined  and  varied,  as  to  pro- 
duce others  of  almost  every  dimension.  And  the  expres- 
sions for  the  tangents  and  secants  may  be  deduced  from  those 
of  the  sines  and  cosines. 


Expressions  for  the  SINE  and  COSINE  of  the  SUM  and  DIFFER- 
ENCE of  arcs. 


207.  Let  o=AH, 
the  greater  of  the 
given  arcs, 

And  &=HL=HD, 
the  less. 

Then  a+6=AH+ 
HL=AL,  the  sum  of 
the  two  arcs, 

And  a — 6=AH — 
HD=AD,  their  differ- 
ence. 


Draw  the  chord  DL,  and  the  radius  CH,  which  may  be 
represented  by  R.  As  DH  is,  by  construction,  equal  to 
HL  ;  DQ  is  equal  to  QL,  and  therefore  DL  is  perpendicular 
to  CH.  (Euc.  3.  3.)  Draw  DO,  HN,  QP,  and  LM,  each 
perpendicular  to  AC  ;  and  DS  and  QB  parallel  to  AC. 

From  the  definitions  of  the  sine  and  cosine,  (Arts.  82,  9.) 
it  is  evident,  that 


TRIGONOMETRICAL    ANALYSIS. 


125 


of  AH,  that  is,  sin  o=»HN, 


The  sine 


of  HL, 
of  AL, 
of  AD, 


sin  6=QL, 
sin  (a+5)=LM, 
sin  (a  —  6)=DO, 


f  of  AH,  that  is,  cos  o=CN, 

The  cosine     ofHL>  cos  6=CQ, 

of  AL,        cos  (a+&)=CM, 

I  of  AD,       cos  (a— 6)=CO. 

The  triangle  CHN  is  obviously  similar  to  CQP ;  and  it  is 
also  similar  to  BLQ,  because  the  sides  of  the  one  are  per- 
pendicular to  those  of  the  other,  each  to  each.  We  have, 
then, 


1.  CH 

CQ 

::HN 

QP,  that 

is,  R 

cos  b  : 

:  sin  a  ;  QP, 

2.  CH 

QL 

::  GST 

BL, 

R 

sin  6  : 

:  cos  a  :  BL, 

3.  CH 

CQ 

::CN 

CP 

R 

cos  b  : 

:  cos  a  :  CP, 

4.  CH 

QL 

::HN 

QB, 

R 

sin  b  : 

:  sin  a  :  QB, 

Converting  each  of  these  proportions  into  an  equation ; 


1.  QP= 


sin  a  cos  b* 


2.  BL 


sin  6  cos  a 


R 


4.  Q] 


cos  a  cos  b 

~~TT~ 

sin  a  sin  b 


R 


Then  adding  the  first  and  second, 

sm  a  cos  ft+sin  b  cos  a 


QP-L.PT, 


R 


Subtracting  the  second  from  the  first, 

sin  a  cos  b  —  sin  b  cos  a 


QP— BL 


R 


*  In  these  formulae,  the  sign  of  multiplication  is  omitted  ;  sin  a  cos  b 
ang  put  for  sin  aXcos  b,  that  is,  the  product  of  the  sine  of  a  into  the 


11* 


TRIGONOMETRICAL   ANALYSIS. 

Subtracting  the  fourth  from  the  third, 

cos  a  cos  b — sin  a  sin  0 


CP— QB=- 


R 


Adding  the  third  and  fourth, 
a  COS  &" 


CP  -f  QB 


R 

But  it  Will  be  seen,  from  the  figure,  that 

QP-f-BL=BM+BL==LM=sin  (a-f-o) 
QP—  BL=QP—  QS=DO=sm  (a—  6) 
CP—  QB=CP—  PM=CM=*=cos  (a+0) 
CP+QB=CP+SD=CO=cos  (a—  b) 

208.  It  then,  for  the  first  member  of  each  of  the  four 
equations  above,  we  substitute  its  value,  we  shall  have, 

sm  a  cos  &+sm  6  cos  a 


I.  sin  (a+b-_ 
ILsii 


-  - 
R 

sin  a  cos  &  —  sin  b  cos  a 


IH,  cos  (a+6)= 


R 

cos  a  cos  6  —  sin  a  sin 


TV          (       K\     cos  a  cos  ^"^"s^n  a  sm 


Or  multiplying  both  sides  by  R, 

R  sin  (a-f&)=sin  a  cos  6-f-sin  b  cos  a 
R  sin  (a  —  6)=sm  a  cos  b  —  sin  b  cos  a 
R  cos  (a+o)=:cos  a  cos  6  —  sin  a  sin  6 
R  cos  (a  —  o)=cos  a  cos  0+  sin  a  sin  o 

That  is,  tfee  product  of  radius  and  the  sine  of  the  sum  of 
two  arcs,  is  equal  to  the  product  of  the  sine  of  the  first  »c 


TRIGONOMETRICAL   ANALYSIS.  127 

into  the  cosine  of  the  second  -f-  the  product  of  the  sine  of 
the  second  into  the  cosine  of  the  first. 

The  product  of  radius  and  the  sine  of  the  difference  of 
two  arcs,  is  equal  to  the  product  of  the  sine  of  the  first  arc 
into  the  cosine  of  the  second  —  the  product  of  the  sine  of 
the  second  into  the  cosine  of  the  first. 

The  product  of  radius  and  the  cosine  of  the  sum  of  tw< 
arcs,  is  equal  to  the  product  of  the  cosines  of  the  arcs  — 
the  product  of  their  sines. 

The  product  of  radius  and  the  cosine  of  the  diference  of 
two  arcs,  is  equal  to  the  product  of  the  cosines  of  the  arcs 
4-  the  product  of  their  sines. 

These  four  equations  may  be  considered  as  fundamental 
propositions,  in  what  is  called  the  Arithmetic  of  Sines  and 
Cosines,  or  Trigonometrical  Analysis. 


Expression  for  the  area  of  a  triangle,  in  terms  of  the  sides. 

209.  Let  the  sides  of  the  triangle 
ABC  be  expressed  by  <L,  6,  and  c,  the 
perpendicular  CD  by  p,  the  seg- 
ment AD  by  d,  and  the  area  by  S. 


Then  ««=£'+<:*—  2«J,  (Euc.  13.  2.) 

Transposing  and  dividing  by  2c  ; 


2c 
By  Euc.  47,  1, 


Reducing  the  fraction,  (Alg.  120.)  and  extracting  the  root 
of  both  sides, 


128  TRIGONOMETRICAL    ANALYSIS. 


V46V—  (62+ca 

This  gives  the  length  of  the  perpendicular  in  terms  of  the 
sides  of  the  triangle.  But  the  area  is  equal  to  the  product 
of  the  base  into  half  the  perpendicular  height.  (Alg.  393.) 
That  is, 


2-f-c2—  a2)2 

Here  we  have  an  expression  for  the  area,  in  terms  of  the 
sides.  But  this  may  be  reduced  to  a  form  much  better 
adapted  to  arithmetical  computation.  It  will  be  seen,  that 
the  quantities  46V2,  and  (62+c2  —  a2)a  are  both  squares  ;  and 
that  the  whole  expression  under  the  radical  sign  is  the  differ- 
ence of  these  squares.  But  the  difference  of  two  squares  is 
equal  to  the  product  of  the  sum  and  difference  of  their  roots. 
(Alg.  191.)  Therefore,  46V—  (62-fc2—  a9)2  may  be  resolved 
into  the  two  factors, 

26c+(62-hc2—  a2)  which  is  equal  to  (6-f  c)2  —  aa 
26c—  (6a+ca—  -of)  which  is  equal  to  a2—  (6—  c)a 

Each  of  these  also,  as  will  be  seen  in  the  expressions  on 
the  right,  is  the  difference  of  two  squares  ;  and  may,  on  the 
same  principle,  be  resolved  into  factors,  so  that, 

{  (b+cy—a2=(b+c+a)X(b+c—a) 
(a+b—  c)x(a—  b+c) 


*  The  expression  for  the  perpendicular  is  the  same,  when  one  of  th« 
angles  is  obtuse,  as  in  Fig.  24.  page  8fr.     Let  A.D=d. 


(Euc.  12,  2.)    And  d- 


Therefore,  ,fc  =Ug.  1C9.) 

v    *          ' 


TRIGONOMETRICAL   ANALYSIS.  129 

Substituting,  then,  these  four  factors,  in  the  place  of  the 
quantity  which  has  been  resolved  into  them,  we  have, 


Here  it  will  be  observed,  that  all  the  three  sides,  a,  6,  and 
c,  are  in  each  of  these  factors. 

Let  h=$(a+b+c)  half  the  sum  of  the  sides.     Then 


X  (h— b)  X 

210.  For  finding  the  area  of  a  triangle,  then,  when  the 
three  sides  are  given,  we  have  this  general  rule ; 

From  half  the  sum  of  the  sides,  subtract  each  side  severally  ; 
multiply  together  the  half  sum  and  the  three  remainders  ;  and 
extract  the  square  root  of  the  product. 


APPLICATION   OF   TRIGONOMETRY 

TO    THE 

i^, MENSURATION      ^ 

OF 

HEIGHTS    AND    DISTANCES. 


ART.  1.  The  most  direct  and  obvious  method  of  deter- 
mining the  distance  or  height  of  any  object,  is  to  apply  to 
it  some  known  measure  of  length,  as  a  foot,  a  yard,  or  a  rod. 
In  this  manner,  the  height  of  a  room  is  found,  by  a  joiner's 
rule  ;  or  the  side  of  a  field  by  a  surveyor's  chain.  But  in 
many  instances,  the  object,  or  a  part,  at  least,  of  the  line 
which  is  to  be  measured,  is  inaccessible.  We  may  wish  to 
determine  the  breadth  of  a  river,  the  height  of  a  cloud,  or 
the  distances  of  the  heavenly  bodies.  In  such  cases  it  is  ne- 
cessary to  measure  some  other  line  ;  from  which  the  required 
line  may  be  obtained,  by  geometrical  construction,  or  more 
exactly,  by  trigonometrical  calculation.  The  line  first  meas- 
ured is  frequently  called  a  base  line. 

2.  In  measuring  angles,  some  instrument  is  used  which 
contains  a  portion  of  a  graduated  circle  divided  into  degrees 
and  minutes.     For  the  proper  measure  of  an  angle  is  an  arc 
of  a  circle,  whose  centre  is  the  angular  point.  (Trig.  74.) 
The  instruments  used  for  this  purpose  are  made  in  different 
forms,  and  with  various   appendages.     The  essential  parts 
are  a  graduated  circle,  and  an  index  with  sight-holes,  for 
taking  the  directions  of  the  lines  which  include  the  angles. 

3.  Angles  of  elevation,  and  of  depression  are  in  a  plane 


MENSURATION    OF    HEIGHTS    AND    DISTANCES. 


131 


perpendicular  to  the  horizon,  which  is  called  a.  vertical  plane* 

An  angle  of  elevation   is  contained  between  a  parallel  to 

the  horizon,  and  an    ascending 

line,  as  BAG.       An   angle    of 

depression  is  contained  between 

a  parallel  to  the  horizon,  and  a 

descending  line,  as  DCA.     The 

complement  of  this  is  the  angle 

ACB. 

4.  The  instrument  by  which  angles  of  elevation,  and  of 
depression,  are  commonly  meas- 
ured, is  called  a   Quadrant.     In 

its  most  simple  form,  it  is  a 
portion  of  a  circular  board  ABC, 
on  which  is  a  graduated  arc  of 
90  degrees,  AB,  a  plumb  line  CP, 
suspended  from  the  central  point 
C,  and  two  sight-holes  D  and  E, 
for  taking  the  direction  of  the 
object. 

To  measure  an  angle  of  elevation  with  this,  hold  the  plane 
of  the  instrument  perpendicular  to  the  horizon,  bring  the 
centre  C  to  the  angular  point,  and  direct  the  edge  AC  in 
such  a  manner,  that  the  object  G  may  be  seen  through  the 
two  sight-holes.  Then  the  arc  BO  measures  the  angle  BCO, 
which  is  equal  to  the  angle  of  elevation  FCG.  For  as  the 
plumb  line  is  perpendicular  to  the  horizon,  the  angle  FCO 
is  a  right  angle,  and  therefore  equal  to  BCG.  Taking  from 
these  the  common  angle  BCF,  there  will  remain  the  angle 
BCO=FCG. 

In  taking  an  angle  of  depression,  as  HCL,  the  eye  is  placed 
at  C,  so  as  to  view  the  object  at  L,  through  the  sight-holes 
D  and  E. 

5.  In  treating  of   the  mensuration  of   heights  and  dis- 


132  MENSURATION    OF 

tances,  no  new  principles  are  to  be  brought  into  view.  We 
have  only  to  make  an  application  of  the  rules  for  the  solu- 
tion of  triangles,  to  the  particular  circumstances  in  which 
the  observer  may  be  placed,  with  respect  to  the  line  to  be 
measured.  These  are  so  numerous,  that  the  subject  may  be 
divided  into  a  great  number  of  distinct  cases.  But  as  they 
are  all  solved  upon  the  same  general  principles,  it  will  not 
be  necessary  to  give  examples  under  each.  The  following 
problems  may  serve  as  a  specimen  of  those  which  most  fre- 
quently occur  in  practice. 

PROBLEM  I. 

TO  FIND  THE  PERPENDICULAR  HEIGHT  OF  AN  ACCESSIBLE  OB- 
JECT STANDING  ON  A  HORIZONTAL  PLANE. 

6.  MEASURE  FROM  THE  OBJECT  TO  A  CONVENIENT  STATION, 
AND  THERE  TAKE  THE  ANGLE  OF  ELEVATION  SUBTENDED  BY 
THE  OBJECT. 

If  the  distance  AB  be  meas- 
ured, and  the  angle  of  elevation 
BAG  ;  there  will  be  given  in 
the  right  angled  triangle  ABC, 
the  base  and  the  angles,  to  find 
the  perpendicular.  (Trig.  137.) 

As  the  instrument  by  which 
the  angle  at  A  is  measured,  is  commonly  raised  a  few  feet 
above  the  ground ;  a  point  B  must  be  taken  in  the  object, 
so  that  AB  shall  be  parallel  to  the  horizon.  The  part  BP, 
may  afterwards  be  added  to  the  height  BC,  found  by  trig- 
onometrical calculation. 

Ex.  1.  What  is  the  height  of  a  tower  BC,  if  the  distance 
AB,  on  a  horizontal  plane,  he  93  feet ;  and  the  angle  BAG 
35i  degrees  ? 

Making  the  hypothenuse  radius  (Tr}g  121.) 


HEIGHTS    AND    DISTANCES.  133 

Cos.  BAG  :  AB  :  :  Sin.  BAG  :  BC=69.9  feet. 

For  the  geometrical  construction  of  the  problem,  see  Trig. 
169. 

2.  What  is  the  height  of  the  perpendicular  sheet  of  water 
at  the  falls  of  Niagara,  if  it  subtends  an  angle  of  40  degrees, 
at  the  distance  of  163  feet  from  the  bottom,  measured  on  a 
horizontal  plane  ?  Ans.  136f  feet. 

7.  If  the  height  of  the  object  be  known,  its  distance  may 
be  found  by  the  angle  of  elevation.  In  this  case  the  angles, 
and  the  perpendicular  of  the  triangle  are  given,  to  find  the 
base. 

Ex.  A  person  on  shore,  taking  an  observation  of  a  ship's 
mast,  which  is  known  to  be  99  feet  high,  finds  the  angle  of 
elevation  3£  degrees.  What  is  the  distance  of  the  ship  from 
the  observer  ?  Ans.  98  rods. 

8.  If  the  observer  be  sta- 
tioned at  the  top  of  the  perpen- 
dicular BC,  whose  height  is 
known  ;  he  may  find  the  length 
of  the  base  line  AB,  by  meas- 
uring the  angle  of  depression 
ACD,  which  is  equal  to  BAG. 

Ex.  A  seaman  at  the  top  of  a  mast  66  feet  high,  looking 
at  another  ship,  finds  the  angle  of  depression  10  degrees. 
What  is  the  distance  of  the  two  vessels  from  each  other  ? 

Ans.  22f  rods. 

We  may  find  the  distance  between  two  objects  which  are 
in  the  same  vertical  plane  with  the  perpendicular,  by  calcu- 
lating the  distance  of  each  from  the  perpendicular.  Thus 
AG  is  equal  to  the  difference  between  AB  and  GB. 

12 


134  MENSURATION    OF 

PROBLEM  II. 

TO    FIND    THE    HEIGHT    OF     AN     ACCESSIBLE     OBJECT    STANDING 

ON  AN  INCLINED  PLANE. 

9.  MEASURE  THE  DISTANCE  FROM  THE  OBJECT  TO  A  CON- 
VENIENT  STATION,    AND    TAKE    THE    ANGLES   WHICH   THIS   BASE 
MAKES  WITH   LINES   DRAWN    FROM   ITS    TWO    ENDS    TO    THE   TOP 
OF    THE    OBJECT. 

If  the  base  AB  be  measured 
and  the  angles  BAG  and  ABC  ; 
there  will  be  given,  in  the  ob- 
lique angled  triangle  ABC,  the 
side  AB,  and  the  angles,  to  find 
BC.  (Trig.  150.)  V'" 

Or  the  height  BC  may   be 

found  by  measuring  the  distances  BA,  AD,  and  taking  the 
angles,  BAG  and  BDC.  There  will  then  be  given  in  the  tri- 
angle ADC,  the  angles  and  the  side  AD,  to  find  AC  ;  and 
consequently,  in  the  triangle  ABC,  the  sides  AB  and  AC 
with  the  angle  BAG,  to  find  BC. 

Ex.  If  AB  be  76  feet,  the  angle  B  101°  25',  and  the  angle 
A  44b  42' ;  what  is  the  height  of  the  tree  BC  ? 

Sin.  C  :  AB  : :  Sin.  A  : :  BC=95.9  feet. 

For  the  geometrical  construction  of  the  problem,  see  Trig. 
169. 

10.  The  following  are  some  of  the  methods  by  which  the 
height  of  an  object  may  be  found,  without  measuring  the 
angle  of  elevation. 

1.  By  shadows.    Let  the  staff  be  be  parallel  to  an  ob* 


HEIGHTS    AND   DISTANCES. 


Ans.  115  feet. 


ject  BC,  whose  height  is  required. 
If  the  shadow  of  BC  extend  to  A,  and 
that  of  be  to  a;  the  rays  of  light  CA 
and  ca  coming  from  the  sun  may  be 
considered  parallel ;  and  therefore  the  tri- 
angles ABC  and  abc  are  similar ;  so  that 

ab  :  be  : :  AB  ;  BC. 

Ex.  If  ab  be  3  feet,  be  5  feet,  and  AB  69  feet,  what  is  the 
height  of  BC  ? 

2.  By  parallel  rods.  If  two 
poles  am  and  en  be  placed 
parallel  to  the  object  BC,  and 
at  such  distances  as  to  bring 
the  points  C,  c,  a,  in  a  line, 
and  if  ab  be  made  parallel  to 
AB  ;  the  triangles  ABC,  and  A 
abc  will  be  similar;  and  we 
shall  have 

ab  :bc::  AB  :  BC. 

One  pole  will  be  sufficient,  if  the  observer  can  place  his 
eye  at  the  point  A,  so  as  to  bring  A,  a,  and  C  in  a  line. 

3.  By  a  mirror.  Let  the  smooth  surface  of  a  body  of 
water  at  A,  or  any  plane  mirror 
parallel  to  the  horizon,  be  so  situ- 
ated, that  the  eye  of  the  observer 
at  c  may  view  the  top  of  the  ob- 
ject C  reflected  from  the  mirror. 
By  a  law  of  Optics,  the  angle  B AC 
is  equal  to  5  Ac  ;  and  if  be  be  made 
parallel  to  BC,  the  triangle  6Ac 
will  be  similar  to  BAG  ;  so  that 

Ab  :  be : :  AB  :  BC. 


186  MENSURATION    OF 

PROBLEM  III. 

TO  FIND  THE  HEIGHT  OF  AN  INACCESSIBLE  OBJECT  ABOVE  4 
HORIZONTAL  PLANE. 

11.  TAKE  TWO  STATIONS  IN   A  VERTICAL  PLANE  PASSING 
THROUGH  THE  TOP  OF  THE    OBJECT,  MEASURE  THE  DISTANCE 
FROM  ONE  STATION  TO  THE  OTHER,  AND  THE  ANGLE  OF  ELE- 
VATION AT  EACH. 

If  the  base  AB  be  measured, 
with  the  angle  GBP  and  CAB  ; 
as  ABC  is  the  supplement  of 
CBP,  there   will  be  given,  in 
the    oblique     angled    triangle 
ABC,  the  side  AB  and  the  an- 
gles, to  find  BC ;  and  then  in 
the  right  angled  triangle  BCP, 
the  hypothenuse  and  the  angles,  to  find  the  perpendicular 
CP. 

Ex.  1.  If  C  be  the  top  of  a  spire,  the  horizontal  base  line 
AB  100  feet,  the  angle  of  elevation  BAG  40°,  and  the  angle 
PBC  60°  ;  what  is  the  perpendicular  height  of  the  spire  ? 

The  difference  between  the  angles  PBC  and  BAG  is  equal 
to  ACB.  (Euc.  32.  1.) 

Then  Sin  ACB  :  AB  : :  Sin  BAG  :  BC=187.9 
And  R  :  BC  : :  Sin  PBC  :  CP=162f  feet. 

2.  If  two  persons  120  rods  from  each  other,  are  standing 
on  a  horizontal  plane,  and  also  in  a  vertical  plane  passing 
through  a  cloud,  both  being  on  the  same  side  of  the  cloud : 
and  if  they  find  the  angles  of  elevation  at  the  two  stations 
to  be  68°  and  76°  ;  what  is  the  height  of  the  cloud  ? 

Ans.  2  miles  135.7  rods. 

12.  The  preceding  problems  are  useful  in  particular  cases. 


HEIGHTS    AND    DISTANCES.  13 Y 

But  the  following  is  a  general  rule,  which  may  be  used  for 
finding  the  height  of  any  object  whatever,  within  moderate 
distances. 

PROBLEM  IV. 

TO    FIND    THE    HEIGHT    OF    ANY    OBJECT,    BY    OBSERVATIONS    AT 

TWO  STATIONS. 

13.  MEASURE  THE  BASE  LINE  BETWEEN  THE  TWO  STATIONS, 
THE  ANGLES  BETWEEN  THIS  BASE  AND  LINES  DRAWN  FROM 

EACH  OF  THE  STATIONS  TO  EACH  END  OF  THE  OBJECT,  AND 
THE  ANGLE  SUBTENDED  BY  THE  OBJECT,  AT  ONE  OF  THE 
STATIONS. 

If  BC  be  the  object 
whose  height  is  requir- 
ed, and  if  the  distance 
between  the  stations  A 
and  D  be  measured,  with 
the  angles  ADC,  DAG, 
ADB,  DAB,  and  BAG  ; 
there  will  be  given  in  the  triangle  ADC,  the  side  AD  and  the 
angles,  to  find  AC  ;  in  the  triangle  ADB,  the  side  AD  and  the 
angles,  to  find  AB ;  and  then,  in  the  triangle  BAG,  the  sides 
AB  and  AC  with  the  included  angle,  to  find  the  required 
height  BC. 

If  the  two  stations  A  and  D  be  in  the  same  plane  with  BC, 
the  angle  BAC  will  be  equal  to  the  difference  between  BAD 
and  CAD.  In  this  case  it  will  not  be  necessary  to  measure 
BAC. 

Ex.  If  AD=83  feet,  (  ADB=33° 

I  ADC=51°  {  DAB=121° 

(  DAG =95°  BAC=26°, 

What  is  the  height  of  the  object  BC  ? 
12* 


138 


MENSURATION    OP 


Sin  ACD  :  AD  : :  ADC  :  AC  =  115.3  (Fig.  8.) 
Sin  ABD  :  AD  : :  ADB  ;  AB= 103.1. 
(AC+AB)  :  (AC— AB)  :  :  Tan  i  (ABC-fACB)  :  Tan  -| 

(ABC— ACB)=13°38' 
Sin  ACB  ;  AB  : :  Sin  BAG  :  BC= 50.57  feet. 

If  the  object  BC  be  perpendicular  to  the  horizon,  its 
height,  after  obtaining  AB  and  AC  as  before,  may  be  found 
by  taking  the  angles  of  elevation  BAP  and  CAP.  The  dif- 
ference of  the  perpendiculars  in  the  right  angled  triangles 
ABP  and  ACP,  will  be  the  height  required. 

PROBLEM  V. 
TO  FIND  THE  DISTANCE  OF  AN  INACCESSIBLE  OBJECT. 

14.  MEASURE  A  BASE  LINE  BETWEEN  TWO  STATIONS,  AND 
THE  ANGLES  BETWEEN  THIS  AND  LINES  DRAWN  FROM  EACH 
OF  THE  STATIONS  TO  THE  OBJECT. 

If  C  be  the  object,  and  if  the 
distance  between  the  stations  A 
and  B  be  measured,  with  the  an- 
gles at  B  and  A ;  there  will  be 
given,  in  the  oblique  angled  tri- 
angle ABC,  the  side  AB  and  the 
angles,  to  find  AC  and  BC,  the 
distances  of  the  object  from  the 
two  stations. 

For  the  geometrical  construction,  see  Trig.  169. 

Ex.  1.  What  are  the  distances  of  the  two  stations  A  and 
B  from  the  house  C,  on  the  opposite  side  of  a  river ;  if 
AB  be  26.6  rods,  B  92°  46',  and  A  38°  40'  ? 

The  angle  C=180— (A+B)=48°  34'.    Then 
Sin  A  :  BC=22.17 


Sin  C  :  AB  : : 


Sin  B  :  AC=35.44. 


HEIGHTS    AND    DISTANCES.  139 

2.  Two  ships  in  a  harbor,  wishing  to  ascertain  how  far 
they  are  from  a  fort  on  shore,  find  that  their  mutual  distance 
is  90  rods,  and  that  the  angles  formed  between  a  line  from 
one  to  the  other,  and  lines  drawn  from  each  to  the  fort  are 
45°  and  56°  15'.  What  are  their  respective  distances  from 
the  fort  ?  Ans.  76.3  and  64.9  rods. 

15.  The  perpendicular  distance  of  the  object  from  the 
line  joining  the  two  stations  may  be  easily  found,  after  the 
distance  from  one  of  the  stations  is  obtained.     The  perpen- 
dicular distance  PC  is  one  of  the  sides  of  the  right  angled 
iriangle  BCP.     Therefore 

R  :  BC  : :  Sin  B  :  PC. 

PROBLEM  VI. 

TO  FIND  THE  DISTANCE  BETWEEN  TWO  OBJECTS,  WHEN  THE 
PASSAGE  FROM  ONE  TO  THE  OTHER,  IN  A  STRAIGHT  LINE 
IS  OBSTRUCTED. 

16.  MEASURE  THE  RIGHT  LINES   FROM  ONE   STATION  TO 

EACH    OF   THE    OBJECTS,  AND   THE    ANGLE    INCLUDED    BETWEEN 

THESE    LINES. 

If  A  and  B  be  the  two  objects, 
and  if  the  distances  BC  and  AC  be 
measured,  with  the  angle  at  C  ;  there 
will  be  given,  in  the  oblique  angled 
triangle  ABC,  two  sides  and  the  in- 
eluded  angle,  to  find  the  other  two 
angles,  and  the  remaining  side.  (Trig. 
153.) 

Ex.  The  passage  between  the  two  objects  A  and  B  being 
obstructed  by  a  morass,  the  line  BC  was  measured  and  found 
to  be  109  rods,  the  line  AC  76  rods,  and  the  angle  at  0 
101°  30'.  What  is  the  distance  AB  ? 

Ans.  144.7  rods. 


140  MENSURATION    OF 

PROBLEM  VII. 
TO  FIND  THE  DISTANCE  BETWEEN  TWO  INACCESSIBLE  OBJECTS. 

17.  MEASURE  A  BASE  LINE  BETWEEN   TWO  STATIONS  AND 
THE  ANGLES  BETWEEN  THIS   BASE  AND   LINES  DRAWN   FROM 
EACH  OF  THE  STATIONS  TO  EACH  OF  THE  OBJECTS. 

If  A  and  B  be  the  two 
objects,  and  if  the  distance 
between  the  stations  C  and  D 
be  measured,  with  the  angles 
BDC,  BCD,  ADC,  and  ACD ; 
the  lines  AC  and  BC  may  be 
found  as  in  Problem  V,  and 
then  the  distance  AB  as  in 
Problem  VI. 

This  rule  is  substantially  the  same  as  that  in  Art.  13. 
The  two  stations  are  supposed  to  be  in  the  same  plane  with 
the  objects.  If  they  are  not,  it  will  be  necessary  to  meas- 
ure the  angle  ACB. 

18.  The  same  process  by  which  we  obtain  the  distance 
of  two  objects  from  each  other,  will  enable  us  to  find  the 
distance  between  one  of  these  and  a  third,  between  that  and 
a  fourth,  and  so  on,  till  a  connection  is  formed  between  a 
great  number  of  remote  points.     This  is  the  plan  of  the 
great  Trigonometrical  Surveys,  which  have  been  lately  car- 
ried on,  with  surprising  exactness,  particularly  in  England 
and  France. 

19.  In  the  preceding  problems  for  determining  altitudes, 
the  objects  are  supposed  to  be  at  such  moderate  distances, 
that  the  observations  are  not  sensibly  affected  by  the  spher- 
ical figure  of  the  earth.  The  height  of  an  object  is  meas- 
ured from  an  horizontal  plane,  passing  through  the  station 
at  which  the  angle  of  elevation  is  taken.  But  in  an  extent 


HEIGHTS    AND    DISTANCES.  141 

of  several  miles,  the  figure  of  the  earth  ought  to  be  taken 
into  account. 

Let  AB  be  a  portion  of  the 
earth's  surface,  H  an  object  above 
it,  and  AT  a  tangent  at  the  point 
A,  or  a  horizontal  line  passing 
through  A.  Then  HT,  the  oblique 
height  of  the  object  above  the  ho- 
rizon of  A,  is  only  zpart  of  the 
height  above  the  surface  of  the 
earth,  or  the  level  of  the  ocean. 
To  obtain  the  true  altitude,  it  is 
necessary  to  add  BT  to  the  height  HT  found  by  obser- 
vation. The  height  BT  may  be  calculated,  if  the  diameter 
of  the  earth  and  the  distance  AT  be  previously  known.  Or 
if  the  height  BT  be  first  determined  from  observation,  witn 
the  distance  AT ;  the  diameter  of  the  earth  may  be  thence 
deduced. 

PROBLEM  VIII. 

TO  FIND  THE  DIAMETER  OF  THE  EARTH,  FROM  THE  KNOWN 
HEIGHT  OF  A  DISTANT  MOUNTAIN,  WHOSE  SUMMIT  IS  JUST 
VISIBLE  IN  THE  HORIZON. 

20.  FROM  THE  SQUARE  OF  THE  DISTANCE  DIVIDED  BY  THE 
HEIGHT,  SUBTRACT  THE  HEIGHT. 

If  BT  (above  figure)  be  a  mountain  whose  height  is 
known,  with  the  distance  AT  ;  and  if  the  summit  T  be  just 
visible  in  the  horizon  at  A ;  then  AT  is  a  tangent  at  the 
point  A. 

Let  2BC=D,  the  diameter  of  the  earth, 
AT=c?,  the  distance  of  the  mountain, 
BT=£,  its  height. 


U2 


MENSURATION    OF 


Then  considering  AT  as  a  straight  line,  and  the  earth  as  a 
sphere,  we  have  (Euc.  36.  3.)* 

(2BC+BT)xBT=ATa;  that  is,  (D+A)Xft=*, 
and  reducing  the  equation, 

— —— -h 
h 

Ex.  The  highest  point  of  the  Andes  is  about  4  miles  above 
the  level  of  the  ocean.  If  a  straight  line  from  this  touch 
the  surface  of  the  water  at  the  distance  of  178-j-  miles  ;  what 
is  the  diameter  of  the  earth  ?  Ans.  7940  miles. 

21.  If  the  distance  AT  be  un- 
known, it  may  be  found  by  meas- 
uring with   a  quadrant  the  angle 
ATC.      Draw   BG   perpendicular 
to  BC;  and  join  CG.     The  trian- 
gles ACG  and  BCG  are  equal,  be- 
cause each  has  a  right  angle,  the 
sides  AC  and  BC  are  equal,  and  the 
hypothenuse  C  G  is  common.  There- 
fore BG  and  AG  are  equal.     In 

the  right  angled  triangle  BGT,  the  angle  BTG  is  given,  and 
the  perpendicular  BT.  From  these  may  be  found  BG  and 
TG,  whose  sum  is  equal  to  AT,  the  distance  required.f 

22.  In  the  common  measurement  of  angles,  the  light  is 
supposed  to  come  from  the  object  to  the  eye  in  a  straight 
.line.     But  this  is  not  strictly  true.     The  direction  of  the 
light  is  affected  by  the  refraction  of  the  atmosphere.     If  the 
object  be  near,  the  deviation  is  very  inconsiderable.     But  in 
an  extent  of  several  miles,  and  particularly  in  such  nice  ob- 

*  Thomson's  Legendre,  30.  4. 

f  Tins  method  of  determining  the  diameter  of  the  earth  is  not  a§  ac- 
curate as  that  by  measuring  a  degree  of  Latitude. 


HEIGHTS    AND    DISTANCES. 


148 


serrations  as  determining  the  height  of  distant  mountains, 
and  the  diameter  of  the  earth,  it  is  necessary  to  make  allow- 
ance for  the  refraction. 

PROBLEM  IX. 

TO  FIND  THE    GREATEST    DISTANCE    AT    WHICH    A    GIVEN  OBJECT 
CAN    BE    SEEN    ON    THE    SURFACE    OF    THE    EARTH. 

23.  To    THE    PRODUCT    OF    THE  HEIGHT  OF  THE    OBJECT  INTO 
THE     DIAMETER     OF     THE     EARTH,    ADD     THE    SQUARE     OF     THE 
HEIGHT  ;    AND    EXTRACT   THE    SQUARE    ROOT    OF    THE    SUM. 

Let  2BC=D,the  diameter  of  the  earth,  (Fig.  12.) 
BT=h,  the  height  of  the  object, 
AT=e?,  the  distance  required. 

Then  (D  +A)  X  h=d\     And  d=Wh+h\ 

Ex.  If  the  diameter  of  the  earth  be  7940  miles,  and 
Mount  JEtna  2  miles  high  ;  how  far  can  its  summit  be  seen 
at  sea  ?  Ans.  126  miles. 

The  actual  distance  at  which  an  object  can  be  seen,  is  in- 
creased by  the  refraction  of  the  air. 

24.  In  this  problem  the  eye  is  supposed  to  be  placed  at 
the  level  of  the  ocean.    But 

if  the  observer  be  elevated 
above  the  surface,  as  on  the 
deck  of  a  ship,  he  can  see 
to  a  greater  distance.  If 
BT  be  the  height  of  the 
object,  and  B'T'  the  height 
of  the  eye  above  the  level 
of  the  ocean ;  the  distance 

\j 

at  which  the  object  can  be 

seen,  is  evidently  equal  to  the  sum  of  the  tangents  AT  and 

AT'. 


144 


MENSURATION    OF 


Ex.  The  top  of  a  ship's  mast  132  feet  high  is  just  visible 
in  the  horizon,  to  an  observer  whose  eye  is  33  feet  above  the 
surface  of  the  water.  What  is  the  distance  of  the  ship  ? 

Ans.  21^-  miles. 

25.  The  distance  to  which  a  person  can  see  the  smooth 
surface  of  the  ocean,  if  no  allowance  be  made  for  refraction, 
is  equal  to  a  tangent  to  the  earth  drawn  from  his  eye,  as 
T'A.  (Fig.  13.) 

Ex.  If  a  man  standing  on  the  level  of  the  ocean,  has  his 
eye  raised  5£  feet  above  the  water :  to  what  distance  can  he 


see  the  surface  ? 

26.  If  the  distance  AT,  with 
the  diameter  of  the  earth  be  given, 
and  the  height  BT  be  required ; 
the  equation  in  Art.  23  gives 


Ans.  2$  miles. 


27.  When  the  diameter  of  the  earth  is  ascertained,  this 
may  be  made  a  base  line  for  determining  the  distance  of  the 
heavenly  bodies.  A  right  angled  triangle  may  be  formed,  the 
perpendicular  sides  of  which  shall  be  the  distance  required, 
and  the  semi-diameter  of  the  earth.  If  then  one  of  the  an- 
gles be  found  by  observation,  the  required  side  may  be  easily 
calculated. 

Let  AC  be  the 
semi-diameter  of 
the  earth,  AH  the 
sensible  horizon  at 
A,  and  CM  the  ra- 
tional horizon,  par- 
allel to  AH,passing 


HEIGHTS    AND    DISTANCES.  145 

through  the  moon  M.  The  angle  HAM  may  be  found  by 
astronomical  observation.  This  angle,  which  is  called  the 
Horizontal  Parallax,  is  equal  to  AMC,  the  angle  at  the  moon 
subtended  by  the  semi-diameter  of  the  earth.  (Euc.  29.  1.) 

PROBLEM  X. 

TO  FIND  THE  DISTANCE  OF    ANY   HEAVENLY   BODY  WHOSE  HOR- 
IZONTAL  PARALLAX   IS   KNOWN. 

28.  AS    RADIUS,  TO    THE  SEMI-DIAMETER  OF  THE  EARTH  J    SO 
IS    THE    CO-TANGENT    OF   THE    HORIZONTAL   PARALLAX,    TO   THE 
DISTANCE. 

In  the  right  angled  triangle  ACM,  (Fig.  14.)  if  AC  be 
made  radius ; 

R  I  AC  : :  Cot.  AMC  ;  CM. 

Ex.  If  the  horizontal  parallax  of  the  moon  be  0°  5*1',  and 
the  diameter  of  the  earth  7940  miles ;  what  is  the  distance 
of  the  moon  from  the  centre  of  the  earth  ? 

Ans.  239,414  miles. 

29.  The  fixed  stars  are  too  far  distant  to  have  any  sensi- 
ble horizontal  parallax.     But  from  late  observations  it  would 
seem,  that  some  of  them  are  near  enough,  to  suffer  a  small 
apparent  change  of  place,  from  the  revolution  of  the  earth 
round  the  sun.     The  distance  of  the  sun,  then,  which  is  the 
semi-diameter  of  the  earth's  orbit,  may  be  taken  as  a  base 
line,  for  finding  the  distance  of  the  stars. 

We  thus  proceed  by  degrees  from  measuring  a  line  on  the 
surface  of  the  earth,  to  calculate  the  distances  of  the 
heavenly  bodies.  From  a  base  line  on  a  plane,  is  deter- 
mined the  height  of  a  mountain  ;  from  the  height  of  a 
mountain,  the  diameter  of  the  earth ;  from  the  diameter  of 
the  earth,  the  distance  of  the  sun,  and  from  the  distance  of 
the  sun  the  distance  of  the  stars. 

13 


146  MENSURATION    OF 

30.  After  finding  the  distance  of  a  heavenly  body,  its  mag- 
nitude is  easily  ascertained ;  if  it  have  an  apparent  diameter, 
sufficiently  large  to  be  measured  by  the  instruments  which 
are  used  for  taking  angles. 

Let  AEB  be  the  an- 
gle which  a  heavenly 
body  subtends  at  the 
eye.  Half  this  angle, 
if  C  be  the  centre  of 
the  body,  is  AEG;  the 
line  EA  is  a  tangent 
to  the  surface,  and  therefore  EAC  is  a  right  angle.  Then 
making  the  distance  EC  radius, 

R  I  EC  : :  Sin.  AEG  :  AC. 

That  is,  radius  is  to  the  distance,  as  the  sine  of  half  the 
angle  which  the  body  subtends,  to  its  semi-diameter. 

Ex.  If  the  sun  subtends  an  angle  of  32'  2' ,  and  if  his 
distance  from  the  earth  be  95  million  miles ;  what  is  his 
diameter?  Ans.  885  thousand  miles. 


PROMISCUOUS  EXAMPLES. 

1.  On  the  bank  of  a  river,  the  angle  of  elevation  of  a  tree 
on  the  opposite  side  is  found  to  be  46°  ;  and  at  another  sta- 
tion 100  feet  directly  back  on  the  same  level,  31°.     What  is 
the  height  of  the  tree  ?  Ans.  143  feet. 

2.  On  a  horizontal  plane,  observations  were  taken  of  a 
tower  standing  on  the  top  of  a  hill.     At  one  station  the  an- 
gle of  elevation  of  the  top  of  the  tower  was  found  to  be  50° ; 
that  at  the  bottom  39°  ;  and  at  another  station  150  feet  di- 
rectly back,  the  angle  of  elevation  of  the  top  of  the  tower 
was  32°.     What  are  the  heights  of  the  hill  and  the  tower  ? 

Ans.  The  hill  is  134  feet  high;  the  tower  63. 


HEIGHTS   AND   DISTANCES.  14T 

3.  What  is  the  altitude  of  the  sun,  when  the  shadow  of  a 
tree,  cast  on  a  horizontal  plane,  is  to  the  height  of  the  tree  as 
4  to  3?  Ans.   36°  62'  12". 

4.  If  a  straight  line  from  the  top  of  the  White  Mountains 
in  New  Hampshire  touch  the  ocean  at  the  distance  of  103-J- 
miles  ?   what  is  the  height  of  the  mountains  ? 

Ans.  7100  feet. 

5.  From  the  top  of  a  perpendicular  rock  55  yards  high, 
the  angle  of  depression  of  the  nearest  bank  of  a  river  is 
found  to  be  55°  54',  that  of  the  opposite  bank  33°  20'.    Re- 
quired the  breadth  of  the  river,  and  the  distance  of  its  near- 
est bank  from  the  bottom  of  the  rock. 

The  breadth  of  the  river  is  46.4  yards  ; 
Its  distance  from  the  rock  37.2. 

6.  If  the  moon  subtend  an  angle  of  31'  14",  when  her  dis- 
tance is  240,000  miles  ;  what  is  her  diameter? 

Ans.  2180  miles. 

7.  Observations  are  made  on  the  altitude  of  a  balloon,  by 
two  persons  standing  on  the  same  side  of  the  balloon,  and  in 
a  vertical  plane  passing  through  it.     The  distance  of  the 
stations  is  half  a  mile.    At  one,  the  angle  of  elevation  is  30° 
58',  at  the  other  36°  52'.     What  is  the  height  of  the  bal- 
loon above  the  ground  ?  Ans.  1-J-  miles. 

8.  The  shadow  of  the  top   of  a  mountain,  when  the  alti- 
tude of  the  sun  on  the  meridian  is  32°,  strikes  a  certain  point 
on  a  level  plain  below ;  but  when  the  meridian  altitude  of 
the  sun  is  67°,  the  shadow  strikes  half  a  mile  farther  south, 
on  the  same  plain.     What  is  the  height  of  the  mountain 
above  the  plain  ?  Ans.  2245  feet. 


NOTES. 


NOTE  A.  p.  13. 

IT  is  common  to  define  logarithms  to  be  a  series  of  numbers 
in  arithmetical  progression,  corresponding  with  another 
series  in  geometrical  progression.  This  is  calculated  to  per- 
plex the  learner,  when,  upon  opening  the  tables,  he  finds  that 
the  natural  numbers,  as  they  stand  there,  instead  of  being  hi 
geometrical,  are  in  arithmetical  progression ;  and  that  the 
logarithms  are  not  in  arithmetical  progression. 

It  is  true,  that  a  geometrical  series  may  be  obtained,  by 
taking  out,  here  and  there,  a  few  of  the  natural  numbers ; 
and  that  the  logarithms  of  these  will  form  an  arithmetical 
series.  But  the  definition  is  not  applicable  to  the  whole  of 
the  numbers  and  logarithms,  as  they  stand  in  the  tables. 

NOTE  B.  p.  89. 

If  the  perpendicular  be  drawn  from  the  angle  opposite 
the  longest  side,  it  will 
always  fall  within  the  tri- 
angle ;  because  the  other 
two  angles  must,  of  course, 
be  acute.  But  if  one  of 
the  angles  at  the  base  be 
obtuse,  the  perpendicular 
will  fall  without  the  trian- 
gle, as  CP. 

In  this  casv,  the  side  on 
which    the    perpendicular 


tfOTES.  149 

falls,  is  to  the  sum  of  the  other  two  ;  as  the  difference  of  the 
latter,  to  the  sum  of  the  segments  made  by  the  perpendicular. 

The  demonstration  is  the  same,  as  in  the  other  case,  ex- 
cept that  AH=BP+PA,  instead  of  BP — PA. 

Thus,  in  the  circle  BDHL,  of  which  C  is  the  centre, 

ABx  AH=ALx  AD  ;  therefore  AB  :  AD  :  :  AL : :  AH. 

But  AD=CD+CA=CB-{-CA 
And  AL=CL— CA=CB— CA 
And  AH=HP+PA=»BP-f  PA 

Therefore, 

AB  :  CB+CA  :  :  CB— CA  :  BP+PA 

When  the  three  sides  are  given,  it  may  be  known  whether 
one  of  the  angles  is  obtuse.  For  any  angle  of  a  triangle  is 
obtuse  or  acute,  according  as  the  square  of  the  side  sub- 
tending the  angle  is  greater,  or  less,  than  the  sum  of  the 
squares  of  the  sides  containing  the  angle.  (Euc.  12,  13.  2.)* 

NOTE  C.  p.  000. 

Gunter's  Sliding  Rule,  is  constructed  upon  the  same  prin- 
ciple as  his  scale,  with  the  addition  of  a  slider,  which  is  so 
contrived  as  to  answer  the  purpose  of  a  pair  of  dividers,  in 
working  proportions,  multiplying,  dividing,  <fec.  The  lines 
on  the  fixed  part  are  the  same  as  on  the  scale.  The  slider 
contains  two  lines  of  numbers,  a  line  of  logarithmic  sines, 
and  a  line  of  logarithmic  tangents. 

To  multiply  by  this,  bring  1  on  the  slider,  against  one  of 
the  factors  on  the  fixed  part ;  and  against  the  other  factor  on 
the  slider,  will  be  the  product  on  the  fixed  part.  To  divide, 
bring  the  divisor  on  the  slider,  against  the  dividend  on  the 
fixed  part ;  and  against  1  on  the  slider,  will  be  the  quotient 

*  Thomson's  Legendre,  12,  13.  4. 


NOTES. 

on  the  fixed  part.  To  work  a  proportion,  bring  the  first  term 
on  the  slider,  against  one  of  the  middle  terms  on  the  fixed 
part ;  and  -against  the  other  middle  term  on  the  slider,  will 
be  the  fourth  term  on  the  fixed  part.  Or  the  first  term 
may  be  taken  on  the  fixed  part ;  and  then  the  fourth  term 
wall  be  found  on  the  slider. 

Another  instrument  frequently  used  in  trigonometrical  con- 
structions, is 

THE   SECTOR. 

This  consists  of  two  equal  scales  movable  about  a  point 
as  a  centre.  The  lines  which  are  drawn  on  it  are  of  two 
kinds,  some  being  parallel  to  the  sides  of  the  instrument,  and 
others  diverging  from  the  central  point,  like  the  radii  of  a  cir- 
cle. The  latter  are  called  the  double  lines,  as  each  is  re- 
peated upon  the  two  scales.  The  single  lines  are  of  the 
same  nature,  and  have  the  same  use,  as  those  which  are  put 
upon  the  common  scale ;  as  the  lines  of  equal  parts,  of 
chords,  of  latitude,  &c.,  on  one  face ;  and  the  logarithmic 
lines  of  numbers,  of  sines,  and  of  tangents,  on  the  other. 

The  double  lines  are 

A  line  of  Lines,  or  equal  parts,  marked    Lin.   or  L. 

A  line  of  Chords,  Cho.  or  C. 

A  line  of  natural  Sines,  Sin.    or  S. 

A  line  of  natural  Tangents  to  45°,  Tan.  or  T. 

A  line  of  tangents,  above  45°,  Tan.  or  T. 

A  line  of  natural  Secants,  Sec.  or  S. 

A  line  of  Polygons,  Pol.  or  P. 

The  double  lines  of  chords,  of  sines,  and  of  tangents  to 
45°,  are  all  of  the  same  radius ;  beginning  at  the  central 
point,  and  terminating  near  the  other  extremity  of  each 
scale ;  the  chords  at  60°,  the  shies  at  90°,  and  the  tangents 
at  45°.  (See  Art.  95.)  The  line  of  lines  is  also  of  the  same 
length,  containing  ten  equal  parts  which  are  numbered,  and 


NOTES.  «         151 

which  are  again  subdivided.  The  radius  of  the  lines  of  se- 
cants and  of  tangents  above  45°,  is  about  one-fourth  of  the 
length  of  the  other  lines.  From  the  end  of  the  radius, 
which  for  the  secants  is  at  0,  and  for  the  tangents  at  45°, 
these  lines  extend  to  between  70°  and  80°.  The  line  of 
polygons  is  numbered  4,  5,  6,  &e.,  from  the  extremity  of 
each  scale,  towards  the  centre. 

The  simple  principle  on  which  the  utility  of  these  several 
pairs  of  lines  depends  is  this,  that  the  sides  of  similar  trian- 
gles are  proportional.  (Euc.  4.  6.)  So  that  sines,  tangents, 
&c.,  are  furnished  to 

A  r* 

any  radius,  within  the 
extent  of  the  opening 
of  the  two  scales.  Let 
AC  and  AC'  be  any 
pair  of  lines  on  the  sec- 
tor, and  AB  and  AB' 
equal  portions  of  these 

lines.  As  AC  and  AC'  are  equal,  the  triangle  AGO7  is 
isosceles,  and  similar  to  ABB'.  Therefore, 

AB  :  AC  :  :  BB'  :  CC'. 

Distances  measured  from  the  centre  on  either  scale,  as  AB 
and  AC,  are  called  lateral  distances.  And  the  distances  be- 
tween corresponding  points  of  the  two  scales,  as  BB'  and 
CC',  are  called  transverse  distances. 

Let  AC  and  CC'  be  radii  of  two  circles.  Then  if  AB  be 
the  chord,  sine,  tangent,  or  secant,  of  any  number  of  degrees 
in  one ;  BB'  will  be  the  chord,  sine,  tangent,  or  secant  of 
the  same  number  of  degrees  in  the  other.  (Art.  119.)  Thus, 
to  find  the  chord  of  80°,  to  a  radius  of  four  inches,  open  the 
sector  so  as  to  make  the  transverse  distance  from  60  to  60, 
on  the  lines  of  chords,  four  inches  ;  and  the  distance  from 
80  to  30,  on  the  same  lines,  will  be  the  chord  required.  To 
find  the  sine  of  28°,  make  the  distance  from  90  to  90,  on  the 


152  KOTES. 

lines  of  sines,  equal  to  radius ;  and  the  distance  from  28  to 
28  will  be  the  sine.  To  find  the  tangent  of  37°,  make  the 
distance  from  45  to  45,  on  the  lines  of  tangents,  equal  to 
radius  ;  and  the  distance  from  37  to  37  will  be  the  tangent. 
In  finding  secants,  the  distance  from  0  to  0  must  be  made 
radius.  (Art.  201.) 

To  lay  down  an  angle  of  34°,  describe  a  circle,  of  any 
convenient  radius,  open  the  sector,  so  that  the  distance  from 
60  to  60  on  the  lines  of  chords  shall  be  equal  to  this  radius, 
and  to  the  circle  apply  a  chord  equal  to  the  distance  from 
34  to  34.  (Art.  161.)  For  an  angle  above  60°,  the  chord 
of  half  the  number  of  degrees  may  be  taken,  and  applied 
twice  on  the  arc,  as  in  Art.  161. 

The  line  of  polygons  contains  the  chords  of  arcs  of  a  cir- 
cle which  is  divided  into  equal  portions.  Thus,  the  distances 
from  the  centre  of  the  sector  to  4,  5,  6,  and  7,  are  the  chords 
°f  •}»  i»  i">  and  -f  of  a  circle.  The  distance  6  is  the  radius. 
(Art.  95.)  This  line  is  used  to  make  a  regular  polygon,  or 
to  inscribe  one  in  a  given  circle.  Thus,  to  make  a  pentagon 
with  the  transverse  distance  from  6  to  6  for  radius,  describe  a 
circle,  and  the  distance  from  5  to  5  will  be  the  length  of  one 
of  the  sides  of  a  pentagon  inscribed  in  that  circle. 

The  line  of  lines  is  used  to  divide  a  line  into  equal  or  pro- 
portional parts,  to  find  fourth  proportionals,  &c.  Thus,  to 
divide  a  line  into  7  equal  parts,  make  the  length  of  the  given 
line  the  transverse  distance  from  7  to  7,  and  the  distance 
from  1  to  1  will  be  one  of  the  parts.  To  find  -f  of  a  line, 
make  the  transverse  distance  from  5  to  5  equal  to  the  given 
line  ;  and  the  distance  from  3  to  3  will  be  ^  of  it. 

In  working  the  proportions  in  trigonometry  on  the  sector, 
the  lengths  of  the  sides  of  triangles  are  taken  from  the  line 
of  lines,  and  the  degrees  and  minutes  from  the  lines  of 
sines,  tangents,  or  secants.  Thus,  in  Art.  135,  ex.  1, 

35  :  R  :  :  26  :  sin  48°. 


NOTES.  153 

To  find  the  fourth  term  of  this  proportion  by  the  sector, 
make  the  lateral  distance  35  on  the  line  of  lines,  a  transverse 
distance  from  90  to  90  on  the  lines  of  sines  ;  then  the  lateral 
distance  26  on  the  line  of  lines,  will  be  the  transverse  dis- 
tance from  48  to  48  on  the  lines  of  shies. 

For  a  more  particular  account  of  the  construction  and  uses 
of  the  Sector,  see  Stone's  edition  of  Bion  on  Mathematical 
Instruments,  Button's  Dictionary,  and  Robertson's  Treatise 
on  Mathematical  Instruments. 


anb   Thomson's   Series. 
A 

PRACTICAL    APPLICATION 

OP 

THE   PRINCIPLES  OF   GEOMETRY 

TO   THE 

MENSURATION 

OP 

SUPERFICIES   AND   SOLIDS. 


ADAPTED  TO 


THE  METHOD  OF  INSTRUCTION  IN  SCHOOLS  AND  ACADEMIES. 


BY  JEREMIAH  DAY,   D.D.  LL.D. 

LATI  PRESIDENT  OP  TALE  COLLEGE. 


NEW   YORK: 

PUBLISHED   BY  MARK  H.  NEWMAN  &  CO., 
No.    199   BROADWAY. 

1848. 


ENTERED,  according  to  Act  of  Congress,  in  the  year  1848,  bj 
JEREMIAH    DAY, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the 
Southern  District  of  New  York. 


THOMAS   B.   SMITH,    STEREOTYPIR, 
316  WILLIAM  STREET,  17.  Y. 


CONTENTS. 


Page 

SECTION    T.  Areas  of  figures  bounded  by  right  lines,      ...  5 

II.  The  Quadrature  of  the  Circle  and  its  parts,  .  19 

Promiscuous  examples  of  Areas, 34 

III.  Solids  bounded  by  plane  surfaces, 37 

IV.  The  Cylinder,  Cone,  and  Sphere, 56 

Promiscuous  examples  of  Solids,      .    ^   .  .  »    .  76 

V.  Isoperimetry,      .    .    ^ 78 

APPENDIX. 

Gauging  of  Casks, 92 

Notes, 99 


SECTION  I. 


AREAS    OP    FIGURES    BOUNDED    BY    RIGHT    LINES. 

ART.  1.  The  following  definitions,  which  are  nearly  the 
same  as  in  Euclid,  are  inserted  here  for  the  convenience  of 
reference. 

I.  Four-sided  figures  have  different  names,  according  to 
the  relative  position  and  length  of  the  sides.  A  parallelo- 
gram has  its  opposite  sides  equal  and  parallel,  as  ABCD. 


(Fig.  2.)  A  rectangle,  or  right  parallelogram,  has  its  oppo- 
site sides  equal,  and  all  its  angles  right  angles ;  as  AC. 
(Fig.  1.)  A  square  has  all  its  sides  equal,  and  all  its  angles 
right  angles ;  as  ABGH.  (Fig.  3.)  A  rhombus  has  all  its 


sides  equal,  and  its  angles  oblique;  as  ABCD.  (Fig.  3.)  A 
rhomboid  has  its  opposite  sides  equal,  and  its  angles  oblique  ; 
as  ABCD.  (Fig.  2.)  A  trapezoid  has  only  two  of  its  sides 
parallel ;  as  ABCD.  (Fig.  4.)  Any  other  four  sided  figure 
is  called  a  trapezium. 


MENSURATION    OF   PLANE   SURFACES. 

II.  A  figure  which  has  more  than  four  sides  is  called  a 
polygon.     A  regular  polygon  has  all  its  sides  equal,  and  all 
its  angles  equal. 

III.  The  height  of  a  triangle 
is    the    length    of    a    perpen- 
dicular, drawn  from  one  of  the 
angles  to  the  opposite  side ;  as 
CP.    The  height  of  a,  four  sided 
figure  is  the  perpendicular  dis- 
tance between  two  of  its  par- 
allel sides ;  as  CP.  (Fig.  4.) 

IV.  The  area  or  superficial  contents  of  a  figure  is  the 
space  contained  within  the  line  or  lines  by  which  the  figure 
is  bounded. 

2.  In  calculating  areas,  some  particular  portion  of  surface 
is  fixed  upon,  as  the  measuring  unit,  with  which  the  given 
figure  is  to  be  compared.     This  is  commonly  a  square  ;  as  a 
square  inch,  a  square  foot,  a  square  rod,  <kc.     For  this  rea- 
son, determining  the  quantity  of  surface  in  a  figure  is  called 
squaring  it,  or  finding  its  quadrature;  that  is,  finding  a 
square  or  number  of  squares  to  which  it  is  equal. 

3.  The  superficial  unit  has  generally  the  same  name,  as 
the  linear  unit  which  forms  the  side  of  the  square. 

The  side  of  a  square  inch  is  a  linear  inch ; 
of  a  square  foot,  a  linear  foot ; 
of  a  square  rod,  a  linear  rod,  &c. 

There  are  some  superficial  measures,  however,  which  have 
no  corresponding  denominations  of  length.  The  acre,  for 
instance,  is  not  a  square  which  has  a  line  of  the  same  name 
for  its  side. 

The  following  tables  contain  the  linear  measures  in  com- 
mon use,  with  their  corresponding  square  measures. 


MENSURATION  OF  PLANE  SURFACES. 


Linear  Measures. 

12     inches  =1  foot. 
3     feet       =1  yard. 
6     feet       =1  fathom. 
16i  feet       =1  rod. 
5£  yards    =1  rod. 
4     rods      =1  chain. 
40     rods      =1  furlong. 

Square  Measures. 

144     inches  =1  foot. 
9     feet       =1  yard. 
36     feet       =1  fathom. 
272-J-  feet       =1  rod. 
30-J-  yards    =1  rod. 
16     rods      =1  chain. 
1600     rods      =1  furlong. 

320     rods      =1  mile. 


102400     rods      =1  mile. 


An  acre  contains  160  square  rods,  or  10  square  chains. 
By  reducing  the  denominations  of  square  measure,  it  will 
be  seen  that 

1  sq.  mile=640  acres=102400  rods=27878400  feet=401 4489600  inches. 
I  acre=10  chains=160  rods=43560  feet=6272640  inches. 

The  fundamental  problem  in  the  mensuration  of  super- 
ficies is  the  very  simple  one  of  determining  the  area  of  a 
right  parallelogram.  The  contents  of  other  figures,  partic- 
ularly those  which  are  rectilinear,  may  be  obtained  by  find- 
ing parallelograms  which  are  equal  to  them,  according  to  the 
principles  laid  down  in  Euclid. 

PROBLEM  I. 

To  find  the  area  of  a  PARALLELOGRAM,  square,  rhombus,  or 
rhomboid. 

4.  MULTIPLY  THE  LENGTH  BY  THE  PERPENDICULAR  HEIGHT 

OR  BREADTH. 

It  is  is  evident  that  the  number  of    D  c 

square  inches  in  the  parallelogram  AC 
is  equal  to  the  number  of  linear 
inches  in  the  length  AB,  repeated  as 
many  times  as  there  are  inches  in  the 
breadth  BC.  For  more  particular 
illustration  of  this  see  Alg.  386 — 389. 


8 


MENSURATION    OF    PLANE    SURFACES. 


The  oblique  parallelogram  or  rhomboid  ABCD,  (Fig.  2.) 
is  equal  to  the  right  parallelogram  GHCD.  (Euc.  36. 1.)*  The 


area,  therefore,  is  equal  to  the  length  AB  multiplied  into  the 
perpendicular  height  HC.  And  the  rhombus  ABCD,  (Fig.  3.) 
is  equal  to  the  parallelogram  ABGH.  As  the  sides  of  a 
square  are  all  equal,  its  area  is  found,  by  multiplying  one  of 
the  sides  into  itself. 

Ex.  1.  How  many  square  feet  are  there  in  a  floor  23£  feet 
long,  and  18  feet  broad  ?  Ans.  23^X18=423. 

2.  What  are  the  contents  of  a  piece  of  ground  which  is 
66  feet  square  ?  Ans.  4356  sq.  feet=16  sq.  rods. 

3.  How  many  square  feet  are  there  in  the  four  sides  of  a 
room  which  is  22  feet  long,  17  feet  broad,  and  11  feet  high? 

Ans.  858. 

ART.  5.  If  the  sides  and  angles  of  a  parallelogram  are 
given,  the  perpendicular  height  may  be  easily  found  by  trig- 
onometry. Thus,  CH  (Fig.  2.)  is  the  perpendicular  of  a 
right  angled  triangle,  of  which  BC  is  the  hypothenuse. 
Then,  (Trig.  134.) 

R  I  BC  :  :  sin  B  ;  CH. 

The  area  is  obtained  by  multiplying  CH  thus  found,  into 
the  length  AB. 


*  Thomson's  Legendre,  i.  5. 


MENSURATION  OF  PLANE  SURFACES. 

Or,  to  reduce  the  two  operations  to  one, 

As  radius, 

To  the  sine  of  any  angle  of  a  parallelogram ; 

So  is  the  product  of  the  sides  including  that  angle, 

To  the  area  of  the  parallelogram. 

For  the  araz=ABxCH,  (Fig.  2.)  But  CH=BC 


R 

Therefore, 

in  B.  Or,R  :  sin  B  :  :  ABxBC  :  thearea. 


Ex.  If  the  side  AB  be  58  rods,  BC  42  rods,  and  the  angle 
B  63°,  what  is  the  area  of  the  parallelogram  ? 

As  radius  10.00000 

To  the  sine  of  B  63°               9.94988 

(  So  is  the  product  of  AB  58                 1.76343 

(  Into  BC  (Trig.  39.)  42                 1.62325 

To  the  area  2170.5  sq.  rods  3.33656 

2.  If  the  side  of  a  rhombus  is  67  feet,  and  one  of  the 
angles  73°,  what  is  the  area  ?  Ans.  4292.7  feet. 

6.  When  the  dimensions  are  given  in  feet  and  inches,  the 
multiplication  may  be  conveniently  performed  by  the  arith- 
metical rule  of  Duodecimals  ;  in  which  each  inferior  denom- 
ination is  one-twelfth  of  the  next  higher.  Considering  a  foot 
as  the  measuring  unit,  a  prime  is  the  twelfth  part  of  a  foot  ; 
a  second,  the  twelfth  part  of  a  prime,  <fec.  It  is  to  be  ob- 
served, that,  in  measures  of  length,  inches  are  primes  ;  but  in 
superficial  measure  they  are  seconds.  In  both,  a  prime  is  iV 
of  a  foot.  But  iV  of  a  square  foot  is  a  parallelogram,  a  foot 
long  and  an  inch  broad.  The  twelfth  part  of  this  is  a  square 
inch,  which  is  TJT  °f  a  square  foot. 
1* 


10  MENSURATION    OF    PLANE    SURFACES. 

Ex.  1»  What  is  the  surface  of  a  board  9  feet  5  inches,  by 
2  feet  7  inches. 

F 

9     5' 
2     7 

18  10 
5     5     11 


24     3     11",  or  24  feet  47  inches. 

2.  How  many  feet  of  glass  are  there  in  a  window  4  feet 
11  inches  high,  and  3  feet  5  inches  broad  ? 

Ans.  16  F.  9'  7",  or  16  feet  115  inches. 

7.  If  the  area  and  one  side  of  a  parallelogram  be  given, 
the  other  side  may  be  found  by  dividing  the  area  by  the 
given  side.  And  if  the  area  of  a  square  be  given,  the  side 
may  be  found  by  extracting  the  square  root  of  the  area.  This 
is  merely  reversing  the  rule  in  Art.  4.  See  Alg.  520,  521. 

Ex.  1.  What  is  the  breadth  of  a  piece  of  cloth  which  is 
36  yds.  long,  and  which  contains  63  square  yards. 

Ans.  If  yds. 

2.  What  is  the  side  of  a  square  piece  of  land  containing 
289  square  rods  ? 

3.  How  many  yards  of  carpeting  If  yard  wide,  will  cover 
a  floor  30  feet  long  and  22£  broad  ? 

Ans.  30X22^  feet=10x7i=75  yds.  And  75-i-H=60. 

4.  What  is  the  side  of  a  square  which  is  equal  to  a  par- 
allelogram 936  feet  long  and  104  broad  ? 

5.  How  many  panes  of  8  by  10  glass  are  there,  in  a  win- 
dow 5  feet  high,  and  2  feet  8  inches  broad  ? 


MENSURATION    OF    PLANE    SURFACES. 


11 


PROBLEM  II. 
To  find  the  area  of  a  TRIANGLE. 

8.  RULE  I.     MULTIPLY  ONE  SIDE   BY  HALF  THE  PERPEN- 
DICULAR FROM  THE  OPPOSITE  ANGLE.    Or,  multiply  half  the 
side  by  the  perpendicular,     Or,  multiply  the  whole  side  by 
the  perpendicular  and  take  half  the  product. 

The  area  of  the  triangle  ABC, 
is  equal  to  %  PCxAB,  because 
a  parallelogram  of  the  same 
base  and  height  is  equal  to  PC 
X  AB,  (Art.  4.)  and  by  Euc.  41, 
1,*  the  triangle  is  half  the  par- 
allelogram. 

Ex.  1.  If  AB  be  65  feet,  and  PC  31.2,  what  is  the  area 
of  the  triangle  ?  Ans.  1014  square  feet. 

2.  What  is  the  surface  of  a  triangular  board,  whose  base 
is  3  feet  2  inches,  and  perpendicular  height  2  feet  9  inches  ? 
.      Ans.  4  F.  4'  3",  or  4  feet  51  inches. 

9.  If  two  sides  of  a  triangle  and  the  included   angle,  are 
given,  the  perpendicular  on  one  of  these  sides  may  be  easily 
found  by  rectangular  trigonometry.     And  the  area  may  be 
calculated  in  the  same  man-      D  c 

ner  as  the  area  of  a  parallel- 
ogram in  Art.  5.  In  the  tri- 
angle ABC, 


R-BC::sinB:CH 


And  because  the  triangle  is  half  the  parallelogram  of  the 
same  base  and  height, 


Thomson's  Legendre,  2.  4. 


1#  .MENSURATION    OF   PLANE   SURFACED 

As  radius, 

To  the  sine  of  any  angle  of  a  triangle  ; 

So  is  the  product  of  the  sides  including  that  angle, 

To  twice  the  area  of  the  triangle.  (Art.  5.) 

Ex.  If  AC  be  39  feet,  AB 
65  feet,  and  the  angle  at  A  53° 
7'  48",  what  is  the  area  of  the 
triangle  ?  Ans.  1014  square  feet. 

9.  6.  If  one  side  and  the  angles 
are  given  ;  then 

As  the  product  of  radius  and  the  sine  of  the  angle  oppo- 
site the  given  side, 

To  the  product  of  the  sines  of  the  two  other  angles ; 
So  is  the  square  of  the  given  side, 
To  twice  the  area  of  the  triangle. 

If  PC  be  perpendicular  to  AB. 

R  t  sin  B  :  :  BC   :  CP 
sin  ACB  :  sin  A  :  :  AB  :  BC 

Therefore,  (Alg.  351,  345.) 

_JK  x  sin  ACB  :  sin  A  X  sin  B  : :  AB  X  BC  :  CP  X  BC  : : 
AB*  :  ABxCP= twice  the  area  of  the  triangle. 

Ex.  If  one  side  of  a  triangle  be  57  feet,  and  the  angles  at 
the  ends  of  this  side  50°  and  60°,  what  is  the  area  ? 

Ans.  1147  sq.  feet. 

10.  If  the  sides  only  of  a  triangle  are  given,  an  angle  may 
be  found,  by  oblique  trigonometry,  Case  IV,  and  then  the 
perpendicular  and  the  area  may  be  calculated.    But  the  area 
may  be  more  directly  obtained,  by  the  following  method. 

RULE  II.  When  the  three  sides  are  given,  from  half  their 
sum  subtract  each  side  severally,  multiply  together  the  half 
sum  and  the  three  remainders,  and  extract  the  square  root  qf 
the  product. 


MENSURATION    OF   PLANE    SURFACES. 


13 


If  the  sides  of  the  triangle  are  a,  b,  and  c,  and  if  A=half 
their  sum,  then 


The  area=Vhx(h—a) X (h—b) X (h—c) 

Ex.  1.  In  the  triangle  ABC, 
given  the  sides  a  52  feet,  b  39, 
and  c  65  ;  to  find  the  side  of  a 
square  which  has  the  same  area 
as  the  triangle. 


h — a=26 


h— 6=39 
h — c=13 


Then  the  area=v78X  26X39X13=1014  square  feet. 
By  logarithms. 


The  half  sum 
First  remainder 
Second    do. 
Third       do. 

le  area  required 

=78 
=  26 
=  39 
=  13 

=  1014 

1.89209 
1.41497 
1.59106 
1.11394 

2)6.01206 
2)3.00603 

Side  of  the  square       =31.843  (Trig.  47.)     1.50301 

2.  If  the  sides  of  a  triangle  are  134,  108,  and  80  rods, 
what  is  the  area?  Ans.  4319. 

3.  What  is  the  area  of  a  triangle  whose  sides  are  371,  264, 
and  225  feet  ? 


PROBLEM  III. 
To  find  the  area  of  a  TRAPEZOID. 

21.    MULTIPLY  HALF  THE    SUM   OF   THE    PARALLEL    SIDES 
INTO  THEIR  PERPENDICULAR  DISTANCE. 


14 


MENSURATION  OF  PLANE  SURFACES. 


The  area  of  the  trapezoid 
ABCD,  is  equal  to  half  the 
sum  of  the  sides  AB  and  CD, 
multiplied  into  the  perpendic- 
ular distance  PC  or  AH.  For 
the  whole  figure  is  made  up  of 
the  two  triangles  ABC  and 

ADC  ;  the  area  of  the  first  of  which  is  equal  to  the  product 
of  half  the  base  AB  into  the  perpendicular  PC,  (Art.  8.) 
and  the  area  of  the  other  is  equal  to  the  product  of  half  the 
base  DC  into  the  perpendicular  AH  or  PC. 

Ex.  If  AB  be  46  feet,  BC  31,  DC  38,  and  the  angle  B 
70°,  what  is  the  area  of  the  trapezoid  ? 

R  ;  BC  :  :  sin  B  :  PC=29.13.    And  42 X  29.13  =  1223*. 

2.  What  are  the  contents  of  a  field  which  has  two  par- 
allel sides  65  and  38  rods,  distant  from  each  other  27  rods  ? 

PROBLEM  IV. 
To  find  the  area  of  a  TRAPEZIUM,  or  of  an  irregular  POLYGON. 

13.  DIVIDE  THE  WHOLE  FIGURE  INTO  TRIANGLES,  BY  DRAW- 
ING DIAGONALS,  AND  FIND  THE  SUM  OF  THE  AREAS  OF  THESE 
TRIANGLES.  (Alg.  394.) 

If  the  perpendiculars  in  two  triangles  fall  upon  the  same 
diagonal,  the  area  of  the  trapezium  formed  of  the  two  trian- 
gles, is  equal  to  half  the 
product  of  the  diagonal  into 
the  sum  of  the  perpendiculars. 

Thus  the  area  of  the  trape- 
zium ABCH,  is 

iBHxAL-KBHxCM=iBHx(AL-fCM.) 
Ex.  In  the  irregular  polygon  ABCDH, 


MENSURATION    OF    PLANE    SURFACES. 


15 


==5.3 


!BH=36  . 

CH~  32'  and  the  perpendiculars  )  CM=9.3 


The  area= 


14.  If  the  diagonals  of  a  trapezium  are  given,  the  area 
may  be  found,  nearly  in  the  same  manner  as  the  area  of  a 
parallelogram  in  Art.  5,  and  the  area  of  a  triangle  in  Art.  9. 

In  the  trapezium  ABCD,  the  sines  of  the  four  angles  at 
N,  the  point  of  intersec- 
tion of  the  diagonals,  are 
all  equal.  For  the  two 
acute  angles  are  supple- 
ments of  the  other  two, 
and  therefore  have  the 
same  sine.  (Trig.  90.) 
Putting,  then,  sin  N  for 

the  sine  of  each  of  these  angles,  the  areas  of  the  four  tri- 
angles of  which  the  trapezium  is  composed,  are  given  by  the 
following  proportions;  (Art.  9.) 


R  :  sin 


- 


2  area  ABN" 
2  area  BCN 
2  area  CDN 
2  area  ADN 


And  by  addition,  (Alg.  349,  Cor.  1.)* 


R  :  8 


2  area  ABCD. 


The  3d  tenn=(AN+CN)x(BN-f-DN)=ACxBD,   by 
the  figure. 

Therefore  R  :  sin  N  :  :  AC  X  BD  :  2  area  ABCD.    That  is, 


*  Euclid,  2,  5.   Gor. 


10  MENSURATION    OF    PLANS    SURFACES. 

As  Radius, 

To  the  sine  of  the  angle  at  the  intersection  of  the 

diagonals  of  a  trapezium  ; 
So  is  the  product  of  the  diagonals, 
To  twice  the  area  of  the  trapezium. 

It  is  evident  that  this  rule  is  applicable  to  a  parallelogram, 
as  well  as  to  a  trapezium. 

If  the  diagonals  intersect  at  right  angles,  the  sine  of  N  is 
equal  to  radius ;  (Trig.  95.)  and  therefore  the  product  of  the 
diagonals  is  equal  to  twice  the  area.  (Alg.  356.)* 

Ex.  1.  If  the  two  diagonals  of  a  trapezium  are  37  and  62, 
and  if  they  intersect  at  an  angle  of  54°,  what  is  the  area  of 
the  trapezium  ?  Ans.  928. 

2.  If  the  diagonals  are  85  and  93,  and  the  angle  of  inter- 
section 74°,  what  is  the  area  of  the  trapezium  ? 


PROBLEM  V. 
To  find  the  area  of  a  REGULAR  POLYGON. 

15.  MULTIPLY  ONE  OF  ITS  SIDES  INTO  HALF  ITS  PERPEN- 
DICULAR DISTANCE  FROM  THE  CENTRE,  AND  THIS  PRODUCT 
INTO  THE  NUMBER  OF  SIDES. 

A  regular  polygon  contains  as  many  equal  triangles  as  the 
figure  has  sides.  Thus,  the  hexagon 
ABDFGH  contains  six  triangles,  each 
equal  to  ABC.  The  area  of  one  of 
them  is  equal  to  the  product  of  the 
side  AB,  into  half  the  perpendicular 
CP.  (Art.  8.)  The  area  of  the  whole, 
therefore,  is  equal  to  this  product 
multiplied  into  the  number  of  sides. 

*  Euclid,  14.  5. 


MENSURATION  OF  PLANE  SURFACES.          17 

Ex.  1.  What  is  the  area  of  a  regular  octagon,  in  which  the 
length  of  a  side  is  60,  and  the  perpendicular  from  the  centre 
72.426?  Ans.  17382. 

2.  What  is  the  area  of  a  regular  decagon  whose  sides  are 
46  each,  and  the  perpendicular  70.7867  ? 

16.  If  only  the  length  and  number  of  sides  of  a  regular 
polygon  be  given,  the  perpendicular  from  the  centre  may  be 
easily  found  by  trigonometry.  The  periphery  of  the  circle 
in  which  the  polygon  is  inscribed,  is  divided  into  as  many 
equal  parts  as  the  polygon  has  sides.  (Euc.  16.4.  Schol.)* 
The  arc,  of  which  one  of  the  sides  is  a  chord,  is  therefore 
known  ;  and  of  course,  the  angle  at  the  centre  subtended  by 
this  arc. 

Let  AB  be  one  side  of  a  regular  polygon  inscribed  in  the 
circle  ABDG.  The  perpendicular  CP  bisects  the  line  AB, 
and  the  angle  ACB.  (Euc.  3.  3.)f  Therefore,  BCP  is  the 
same  part  of  360°,  which  BP  is  of  the  perimeter  of  the 
polygon.  Then,  in  the  right  angled  triangle  BCP,  if  BP  be 
radius,  (Trig.  122.) 

R  :  BP  :  :  cot  BCP  :  CP.     That  is, 

As  Radius, 

To  half  of  one  of  the  sides  of  the  polygon  ; 
So  is  the  cotangent  of  the  opposite  angle, 
To  the  perpendicular  from  the  centre. 

Ex.  1.  If  the  side  of  a  regular  hexagon  be  38  inches,  what 
is  the  area  ? 


The  angle  BCP^  of  360°=30°.     Then, 
R  :  19  :  :  cot  30°  ;  32.909=CP,  the  perpendicular, 
And  the  area=19X32.909X6  = 


*  Thomson's  Legendre,  2.  5.  Schol.  f  ^id.  6.  2. 


18 


MENSURATION    OF    PLANE    SURFACES. 


2.  What  is  the  area  of  a  regular  decagon  whose  sides  are 
each  62  feet  ?  Ans.  29576. 

17.  From  the  proportion  in  the  preceding  article,  a  table 
of  perpendiculars  and  areas  may  be  easily  formed,  for  a  series 
of  polygons,  of  which  each  side  is  a  unit.  Putting  R=l, 
(Trig.  100.)  and  w=the  number  of  sides,  the  proportion  be- 
comes 

Q  Af)° 

1  ;  £  :  :  cot ;  the  perpendicular 


So  that,  the  perp.=%  co 


360 


And  the  area  is  equal  to  half  the  product  of  the  perpen- 
dicular into  the  number  of  sides.  (Art.  15.) 

Thus,  in  the  trigon,  or  equilateral  triangle,  the  perpendic- 

QfiftO 

ular=i  cot  =j  cot  60°=0.2886752. 


G 


And  the  area= 0.4330127. 


In  the  tetragon,  or  square,  the  perpendicular =£  co 


.360° 
IT 


=•£  cot  45°=0.5.     And  the  area=l. 

In  this  manner,  the  following  table  is  formed,  in  which 
the  side  of  each  polygon  is  supposed  to  be  a  unit. 


A  TABLE  OF  REGULAR  POLYGONS. 


Names.                 |  Sides,  j  Angles. 

Perpendiculars. 

Areas. 

Trigon, 

3 

60° 

,0.2886752 

0.4330127 

Tetragron, 

4 

45° 

0.5000000 

1.0000000 

Pentagon, 

5 

36° 

0.6881910 

1.7204774 

Hexagon, 

6 

30° 

0.8660254 

2.5980762 

Heptagon, 

7 

25* 

1.0382601 

3.6339124 

Octagon, 

8 

22£ 

1.2071069 

4.8284271 

Nonagon, 

9 

20° 

1.3737385 

6.1818242 

Decagon, 

10 

18° 

1.5388418 

7.6942088 

Undecagon, 

11 

16^r 

1.7028439 

9.3656399 

Dodecagon, 

12 

15° 

1.8660252 

11.1961524 

MENSURATION    OP   THE    CIRCLE.  19 

By  this  table  may  be  calculated  the  area  of  any  other  reg- 
ular polygon,  of  the  same  number  of  sides  with  one  of  these. 
For  the  areas  of  similar  polygons  are  as  the  squares  of  their 
homologous  sides.  (Euc.  20.  6.)* 

To  find,  then,  the  area  of  a  regular  polygon,  multiply  the 
square  of  one  of  its  sides  by  the  area  of  a  similar  polygon  of 
which  the  side  is  a  unit. 

Ex.  1.  What  is  the  area  of  a  regular  decagon  whose  sides 
are  each  102  rods  ?  Ans.  80050.5  rods. 

2.  What  is  the  area  of  a  regular  dodecagon  whose  sides 
are  each  87  feet  ? 


SECTION  II. 

THE    QUADRATURE    OF    THE    CIRCLE    AND    ITS    PARTS. 

ART.  18.  Definition  I.  A  circle  is  a  plane  bounded  by  a 
line  which  is  equally  distant  in  all  its  parts  from  a  point 
within  called  the  centre.  The  bounding  line  is  called  the 
circumference  or  periphery.  An  arc  is  any  portion  of  the 
circumference.  A  semi-circle  is  half,  and  a  quadrant  one- 
fourth  of  a  circle. 

II.  A  Diameter  of  a  circle  is  a  straight  line  drawn  through 
the  centre,  and  terminated  both  ways  by  the  circumference. 
A  Radius  is  a  straight  line  extending  from  the  centre  to  the 
circumference.     A  Chord  is  a  straight  line  which  joins  the 
two  extremities  of  an  arc. 

III.  A  Circular  Sector  is  a  space  contained  between  an 
arc  and  the  two  radii  drawn  from  the  extremities  of  the  arc. 

*  Thomson's  Legendre  1.  5.  Cor. 


20 


MENSURATION    OF   THE    CIRCLE. 


It  may  be  less  than  a  semi-circle,  as 
ACBO,  or  greater,  as  ACBD. 

IV.  A  Circular  Segment  is  the 
space  contained  between  an  arc  and 
its  chord,  as  ABO  or  ABD.  The  chord 
is  sometimes  called  the  base  of  the 
segment.  The  height  of  a  segment 
is  the  perpendicular  from  the  middle 
of  the  base  to  the  arc,  as  PO. 

V»  A  Circular  Zone  is  the  space 
between  two  parallel  chords,  as 
AGHB.  It  is  called  the  middle 
zone,  when  the  two  chords  are 
equal,  as  GHDE. 


VI.  A  Circular  Ring  is  the  space  between  the  peripheries 
of  two  concentric  circles,  as  AA',  BB'.  (Fig.  13.) 


VII.  A  Lune  or  Crescent  is  the  space  between  two  circu- 
lar arcs  which  intersect  each  other,  as  ACBD.  (Fig.  14.) 

19.  The  Squaring  of  the  Circle  is  a  problem  which  has 
exercised  the  ingenuity  of  distinguished  mathematicians  for 
many  centuries.  The  result  of  their  efforts  has  been  only 
an  approximation  to  the  value  of  the  area.  This  can  be  car* 
ried  to  a  degree  of  exactness  far  beyond  what  is  necessary 
for  practical  purposes. 


MENSURATION    OF   THE    CIRCLE.  21 

20.  If   the  circumference  of  a  circle  of  given  diameter 
were  known,  its  area  could  be  easily  found.     For  the  area  is 
equal  to  the  product  of  half  the  circumference  into  half  the 
diameter.  (Sup.  Euc.  5,  l.*)f     But  the  circumference  of  a 
circle  has  never  been  exactly  determined.     The  method  of 
approximating  to  it  is  by  inscribing  and  circumscribing  poly- 
gons, or  by  some  process  of  calculation  which  is,  in  principle, 
the  same.     The  perimeters  of  the  polygons  can  be  easily 
and  exactly  determined.      That  which   is  circumscribed  is 
greater,  and  that  which  is  inscribed  is  less,  than  the  peri- 
phery of  the  circle ;  and  by  increasing  the  number  of  sides, 
the  difference  of  the  two  polygons  may  be  made  less  than 
any  given  quantity.  (Sup.  Euc.  4,  1.) 

21.  The  side  of  a   liexagon  in- 
scribed  in  a  circle,  as  AB,  is  the 
chord  of  an  arc  of  60°,  and  there- 
fore equal  to  the  radius.  (Trig.  95.) 
The  chord  of  half  this  arc,  as  BO, 
is  the  side  of  a  polygon  of  12  equal 
sides.     By  repeatedly  bisecting  the 
arc,  and  finding  the  chord,  we  may 

obtain  the  side  of  a  polygon  of  an  immense  number  of  sides. 
Or  we  may  calculate  the  sine,  which  will  be  half  the  chord 
of  double  the  arc,  (Trig.  82,  cor.,)  and  the  tangent,  which 
will  be  half  the  side  of  a  similar  circumscribed  polygon. 
Thus  the  sine  AP,  is  half  of  AB,  a  side  of  the  inscribed 
hexagon ;  and  the  tangent  NO  is  half  of  NT,  a  side  of  the 
circumscribed  hexagon.  The  difference  between  the  sine 
and  the  arc  AO  is  less  than  the  difference  between  the  sine 
and  the  tangent.  In  the  section  on  the  computation  of  the 
canon,  (Trig.  223.)  by  12  successive  bisections,  beginning 
with  60  degrees,  an  arc  is  obtained  which  is  the  irrSTT  °f 
the  whole  circumference. 

*  In  this  manner,  the  Supplement  to  Play  fair's  Euclid  is  referred  t&  in 
this  work.  t  Thomson's  Legendre,  11.  5. 


22  MENSURATION    OF   THE    CIRCLE. 

The  cosine  of  this,  if  radius  be  1,  is  found  to  be  .99999996732 
The  sine  is  .00025566346 

And  the  tangent—  sme  (Trig.  93.)             =.00025566347 
cosine  

The  diff.  between  the  sine  and  tangent  is  only  .00000000001 
And  the  difference  between  the  sine  and  the  arc  is  still  less. 

Taking  then,  .000255663465  for  the  length  of  the  arc, 
multiplying  by  24576,  and  retaining  8  places  of  decimals, 
we  have  6.28318531  for  the  whole  circumference,  the  radius 
being  1.  Half  of  this, 

3.14159265 

is  the  circumference  of  a  circle  whose  radius  is  £,  and  diam- 
eter 1. 

22.  If  this  be  multiplied  by  7,  the  product  is  21. 99 -for 
22  nearly.  So  that, 

Diam  :  Circum  :  :  7  :  22,  nearly. 

If  3.14159265  be  multiplied  by  113,  the  product  is 
354.9999+,  or  355,  very  nearly.  So  that, 

Diam  ;  Circum  :  :  113  ;  355,  very  nearly. 

The  first  of  these  ratios  was  demonstrated  by  Archimedes. 

There  are  various  methods,  principally  by  infinite  series 
and  fluxions,  by  which  the  labor  of  carrying  on  the  approx- 
imation to  the  periphery  of  a  circle  may  be  very  much 
abridged.  The  calculation  has  been  extended  to  nearly  150 
places  of  decimals.  But  four  or  five  places  are  sufficient 
for  most  practical  purposes. 

After  determining  the  ratio  between  the  diameter  and  the 
circumference  of  a  circle,  the  following  problems  are  easily 
solved. 


MENSURATION    OF   THE    CIRCLE.  23 


PROBLEM  I. 
To  find  the  CIRCUMFERENCE  of  a  circle  from  its  diameter"  t 

23.  MULTIPLY  THE  DIAMETER  BY  3*14159.* 

Or, 

Multiply  the  diameter  by  22  and  divide  the  product  by  1.  Or, 
multiply  the  diameter  by  355,  and  divide  the  product  by 
113.  (Art.  22.) 

Ex.  1.  If  the  diameter  of  the  earth  be  7930  miles,  what 
is  the  circumference?  Ans.  249128  miles. 

2.  How  many  miles  does  the  earth  move,  in  revolving 
round  the  sun ;  supposing  the  orbit  to  be  a  circle  whose 
diameter  is  190  million  miles  ?  Ans.  596,902,100. 

3.  What  is  the  circumference  of  a  circle  whose  diameter 
is  769843  rods  ? 

PROBLEM  II. 
To  find  the  DIAMETER  of  a  circle  from  its  circumference. 

24.  DIVIDE  THE  CIRCUMFERENCE  BY  3. 14 159* 

Or, 

Multiply  the  circumference  by  7,  and  divide  the  product  by 
22.  Or,  multiply  the  circumference  by  113,  and  divide 
the  product  by  355.  (Art.  22.) 

Ex.  1.  If  the  circumference  of  the  sun  be  2,800,000  miles, 
what  is  his  diameter?  Ans.  891,267. 

2.  What  is  the  diameter  of  a  tree  which  is  5£  feet  round  ? 

25.  As  multiplication  is  more  easily  performed  than  divis- 
ion, there  will  be  an  advantage  in  exchanging  the  divisor 

*  In  many  cases,  3.1416  will  be  sufficiently  accurate. 


24  MENSURATION  OF  THE  CIRCLE. 

3.14159  for  a  multiplier  which  will  give  the  same  result. 
In  the  proportion 

3.14159  :  1  :  :  Circum  :  Diam. 

to  find  the  fourth  term,  we  may  divide  the  second  by  the 
first,  and  multiply  the  quotient  into  the  third.  Now,  1~ 
3.14159=0.31831.  If,  then,  the  circumference  of  a  circle 
be  multiplied  by  .31831,  the  product  will  be  the  diameter. 

Ex.  1.  If  the  circumference  of  the  moon  be  6850  miles, 
what  is  her  diameter  ?  Ans.   2180. 

2.  If  the  whole  extent  of  the  orbit  of  Saturn  be  5650 
million  miles,  how  far  is  he  from  the  sun  ? 

3.  If  the  periphery  of  a  wheel  be  4  feet  7  inches,  what  is 
its  diameter? 


PROBLEM  III. 
To  find  the  length  of  an  ARC  of  a  circle. 

26.  As  360°,  to  the  number  of  degrees  in  the  arc  ; 

So  is  the  circumference  of  the  circle,  to  the  length  of  the  arc. 

The  circumference  of  a  circle  being  divided  into  360°, 
(Trig.  73.)  it  is  evident  that  the  length  of  an  arc  of  any  less 
number  of  degrees  must  be  a  proportional  part  of  the  whole. 

Ex.  What  is  the  length  of  an  arc  of  16°,  in  a  circle  whose 
radius  is  50  feet  ? 

The  circumference  of  the  circle  is  314.159  feet.  (Art.  23.) 
Then  360  :  16  :  :  314.159  :  13.96  feet. 

2.  If  we  are  95  millions  of  miles  from  the  sun,  and  if  the 
earth  revolves  round  it  in  365^  days,  how  far  are  we  carried 
in  24  hours  ?  Ans.  1  million  634  thousand  miles. 

27.  The  length  of  an  arc  may  also  be  found,  by  multiply- 
ing the  diameter  into  the  number  of  degrees  in  the  arc,  and 


MENSURATION    OF    THE    CIRCLE. 


25 


this  product  into  .0087266,  which  is  the  length  of  one  de- 
gree, in  a  circle  whose  diameter  is  1.  For  3. 141 59 -f- 360= 
0.00872  66.  And  in  different  circles,  the  circumferences,  and 
of  course  the  degrees,  are  as  the  diameters.  (Sup.  Euc.  8,  1.)* 

Ex.  1.  What  is  the  length  of  an  arc  of  10°  15'  in  a  circle 
whose  radius  is  68  rods  ?  Ans.  12.165  rods. 

2.  If  the  circumference  of  the  earth  be  24913  miles,  what 
is  the  length  of  a  degree  at  the  equator  ? 

28.  The  length  of  an  arc  is  frequently  required,  when 
the  number  of  degrees  is  not  given.  But  if  the  radius  of  the 
circle,  and  either  the  chord  or  the  height  of  the  arc,  be 
known  ;  the  number  of  degrees 
may  be  easily  found. 

Let  AB  be  the  chord,  and  PO 
the  height,  of  the  arc  AOB,  As 
the  angles  at  P  are  right  angles,  and 
AP  is  equal  to  BP;  (Art.  18.  Def. 
4.)  AO  is  equal  to  BO.  (Euc.  4, 
l.)f  Then, 

BP  is  the  sine,  and  CP  the  cosine, 


OP  the  versed  sine,  and  BO  the  chord, 


of  half  the  arc  AOB. 


CB  :  R  :  : 


And  in  the  right  angled  triangle  CBP, 

BP  :  sin  BCP  or  BO. 
CP  :  cos  BCP  or  BO. 

Ex.  1.  If  the  radius  C0=25,  and  the  chord  AB=43.3; 
what  is  the  length  of  the  arc  AOB  ? 

CB  :  R  :  :  BP  :  sin  BCP  or  B0=60°  very  nearly. 
The  circumference  of  the  circle  =3.14159X50=157.08. 
And  360°  :  60°  : :  157.08  : 26.1 8=OB.  Therefore,AOB=52.36, 


*  Thomson's  Legendre,  10.  5. 


t  Ibid.,  5.  1. 


20  MENSURATION    OF    THE    CIRCLE. 

2.  What  is  the  length  of  an  arc  whose  chord  is  21 6^-,  in  a 
circle  whose  radius  is  125  ?  Ans.  261.8. 

20.  If  only  the  chord  and  the  height  of  an  arc  be  given, 
the  radius  of    the   circle   may   be 
found,  and  then  the  length  of  the 
arc. 


If  BA  be  the  chord,  and  PO  the 
the  height  of  the  arc  AOB,  then 
(Euc.  35.  3.)* 


DP 


—  .      And  DO=OP+DP==OP+g£l. 


That  is,  the  diameter  is  equal  to  the  height  of  the  arc,  + 
the  square  of  half  the  chord  divided  by  the  height. 

The  diameter  being  found,  the  length  of  the  arc  may  be 
calculated  by  the  two  preceding  articles. 

Ex.  1.  If  the  chord  of  an  arc  be  173.2,  and  the  height  50, 
what  is  the  length  of  the  arc  ? 


The   diameter  =50 -f^?!=200.   The  arc  contains  120°, 
50 

(Art.  28.)  and  its  length  is  209.44.  (Art.  26.) 

2.  What  is  the  length  of  an  arc  whose  chord  is  120,  and 
height  45  ?  Ans.   160.8. 

PROBLEM  IV. 
To  find  the  AREA  of  a  CIRCLE. 

30.  MULTIPLY    THE    SQUARE   OF   THE   DIAMETER    BY  TUB 
DECIMALS  .7854* 


*  Thomson's  Legendre,  10.  5. 


MENSURATION  OF  THE  CIRCLE.  27 

Or, 

MULTIPLY  HALF  THE  DIAMETER  INTO  HALF  THE  CIRCUM- 
FERENCE. Or,  multiply  the  whole  diameter  into  the  whole 
circumference,  and  take  -J-  of  the  product. 

The  area  of  a  circle  is  equal  to  the  product  of  half  the 
diameter  into  half  the  circumference  ;  (Sup.  Euc.  5,  1.)  or, 
which  is  the  same  thing,  \  the  product  of  the  diameter  and 
circumference.  If  the  diameter  be  1,  the  circumference  is 
3.14159;  (Art.  23.)  one-fourth  of  which  is  0.7854  nearly. 
But  the  areas  of  different  circles  are  to  each  other,  as  the 
squares  of  their  diameters.  (Sup.  Euc.  7,  1.)*  The  area  of 
any  circle,  therefore,  is  equal  to  the  product  of  the  square  of 
its  diameter  into  0.7854,  which  is  the  area  of  a  circle  whose 
diameter  is  1. 

Ex.  1.  What  is  the  area  of  a  circle  whose  diameter  is  623 
feet?  Ans.  304836  square  feet. 

2.  How  many  acres  are  there  in  a  circular  island  whose 
diameter  is  124  rods.  Ans.    75  acres,  and  76  rods. 

3.  If  the  diameter  of  a  circle  be  113,  and  the  circumfer- 
ence 355,  what  is  the  area?  Ans.  10029. 

4.  How  many  square  yards  are  there  in  a  circle  whose 
diameter  is  7  feet  ? 

31.  If  the  circumference  of  a  circle  be  given,  the  area  may 
be  obtained,  by  first  finding  the  diameter ;  or,  without  finding 
the  diameter,  by  multiplying  the  square  of  the  circumference 
by  .07958. 

For,  if  the  circumference  of  a  circle  be  1,  the  diameter  = 
1-7-3.14159=0.31831  ;  and  £  the  product  of  this  into  the 
circumference  is  .07958  the  area.  But  the  areas  of  different 
circles,  being  as  the  squares  of  their  diameters,  are  also  as 
the  squares  of  their  circumferences.  (Sup.  Euc.  8,  1.) 

*  Thomson's  Legendre,  28.  4.   Cor. 


28 


MENSURATION    OF   THE    CIRCLE, 


Ex.  1.  If  the  circumference  of  a  circle  be  136  feet,  what 
is  the  area?  Ans.  1472  feet. 

2.  What  is  the  surface  of  a  circular  fish-pond,  which  is 
10  rods  in  circumference  ? 

32.  If  the  area  of  a  circle  be  given,  the  diameter  may  be 
found,  by  dividing  the  area  by  .7854,  and  extracting  the 
square  root  of  the  quotient. 

This  is  reversing  the  rule  in  Art.  30. 

Ex.  1.  What  is  the  diameter  of  a  circle  whose  area  is 
380.1336  feet  ? 

Ans.  380.1336-r-.7854=484.     And  V484=22. 

2.  What  is  the  diameter  of  a  circle  whose  area  is  19.635  ? 

33.  The  area  of  a  circle,  is  to  the  area  of  the  circumscribed 
square  /  as  .7854  to  1  ;  and  to  that  of  the  inscribed  square 
as  .7854  to  |. 

Let  ABDF  be  the  inscribed 
square,  and  LMNO  the  circum- 
scribed square,  of  the  circle  ABDF. 
The  area  of  the  circle  is  equal  to 
AF2X-7854.  (Art.  30.)  But  the  A 
area  of  the  circumscribed  square 

(Art.  4.)  is  equal  to  ON*=ADl 
And  the  smaller  square  is  half  of 
the  larger  one.  For  the  latter  con- 
tains 8  equal  triangles,  of  which  the  former  contains  only  4. 

Ex.  What  is  the  area  of  a  square  inscribed  in  a  circle 
whose  area  is  159  ?  Ans.  .7854  :  i  :  :  159  :  101.22. 

PROBLEM  V. 
To  find  the  area  of  a  SECTOR  of  a  circle. 

34.  MULTIPLY  THE    RADIUS  INTO   HALF   THE  LENGTH    OF 

THE    ARC. 


MENSURATION    OF   THE    CIRCLE. 


29 


Or, 

As  360,  TO  THE  NUMBER  OF  DEGREES  IN  THE  ARC  J 
So  IS  THE  AREA  OF  THE  CIRCLE,  TO  THE  AREA  OF  THE 
SECTOR. 

It  is  evident,  that  the  area  of  the  sector  has  the  same 
ratio  to  the  area  of  the  circle,  which  the  length  of  the  arc 
has  to  the  length  of  the  whole  circumference ;  or  which  the 
number  of  degrees  in  the  arc  has  to  the  number  of  degrees 
in  the  circumference. 

Ex.  1.  If  the  arc  AOB  be  120°, 
and  the  diameter  of  the  circle  226  ; 
what  is  the  area  of  the  sector 
AOBC? 

The  area  of  the  whole  circle   is 
40115.  (Art.  30.) 

And   360°    :    120°  :  :  40115    : 
133711,  the  area  of  the  sector. 

2.  What  is  the  area  of  a  quadrant  whose  radius  is  621  ? 

3.  What  is  the  area  of  a  semi- circle,  whose  diameter  is 
328? 

4.  What  is  the  area  of  a  sector  which  is  less  than  a  semi- 
circle, if  the  radius  be  15,  and  the  chord  of  its  arc  12  ? 

Half  the  chord  is  the  sine  of  23°  34f '  nearly.  (Art.  28.) 

The  whole  arc,  then,  is  47°    9£' 

The  area  of  the  circle  is          706.86 

And  360°  :  47°  9£' : :  706.86  :  92.6  the  area  of  the  sector. 

5.  If  the  arc  ABB  be  240  degrees,  and  the  radius  of  the 
circle  113,  what  is  the  area  of  the  sector  ADBC  ? 

PROBLEM  VI. 

To  find  the  area  of  a  SEGMENT  of  a  circle. 
35.    FIND   THE    AREA  OF    THE    SECTOR  WHICH    HAS   THE 


80  MENSURATION    Of   THE    CIRCLE. 

SAME  ARC,  AND  ALSO  THE  AREA  OF  A  TRIANGLE  FORMED  BT 
THE  CHORD  OF  THE  SEGMENT  AND  THE  RADII  OF  THE  SEC- 
TOR. 

THEN,  IF  THE  SEGMENT  BE  LESS  THAN  A  SEMI-CIRCLE; 
SUBTRACT  THE  AREA  OF  THE  TRIANGLE  FROM  THE  AREA  OF 
THE  SECTOR.  BUT,  IF  THE  SEGMENT  BE  GREATER  THAN  A 

SEMI-CIRCLE,  ADD  THE  AREA  OF  THE  TRIANGLE  TO  THE  AREA 
OF  THE  SECTOR. 

If  the  triangle  ABC,  be  taken 
from  the  sector  AOBC,  it  is  evi- 
dent the  difference  will  be  the  seg- 
ment AOBP,  less  than  a  semi-cir- 
cle. And  if  the  same  triangle  be 
added  to  the  sector  ADBC,  the 
sum  will  be  the  segment  ADBP, 
greater  than  a  semi-circle. 

The  area  of  the  triangle  (Art.  8.) 

is  equal  to  the  product  of  half  the  chord  AB  into  CP,  which 
is  the  difference  between  the  radius  and  PO  the  height  of  the 
segment.  Or  CP  is  the  cosine  of  half  the  arc  BOA.  If  this 
cosine  and  the  chord  of  the  segment  are  not  given,  they 
may  be  found  from  the  arc  and  the  radius. 

Ex.  1.  If  the  arc  AOB  be  120°,  and  the  radius  of  the 
circle  be  113  feet,  what  is  the  area  of  the  segment  AOBP  ? 

In  the  right  angled  triangle  BCP, 
R  :  BC  :  :  sin  BCO  :  BP=97.86,  half  the  chord.  (Art.  28.) 

The  cosine  PC=i  CO  (Trig.  96,  Cor.)  =56.5 

The  area  of  the  sector  AOBC  (Art.  34.)  =13371.67 

The  area  of  the  triangle  ABC=BPxPC  =  5528.97 

The  area  of  the  segment,  therefore,  =   7842.7 


2.  If  the  base  of  a  segment,  less  than  a  semi-circle,  be  10 


MENSURATION    OF   THE    CIRCLE.  31 

feet,  and  the  radius  of  the  circle   12  feet,  what  is  the  area 
of  the  segment  ? 

The  arc  of  the  segment  contains     49-}-     degrees.  (Art.  28.) 
The  area  of  the  sector  =61.89  •    (Art.  34.) 

The  area  of  the  triangle  =54.54 

And  the  area  of  the  segment      =  7.35  square  feet. 

3.  What  is  the  area  of  a  circular  segment,  whose  height 
is  19.2  and  base  70  ?  Ans.  947.86. 

4.  What  is  the  area  of  the  segment  ADBP,  (Fig.  9.)  if 
the  base  AB  be  195.7,  and  the  height  PD  169.5  ? 

Ans.  32272. 

36.  The  area  of  any  figure  which  is  bounded  partly  by 
arcs  of  circles,  and  partly  by  right  lines,  may  be  calculated, 
by  finding  the  areas  of  the  segments  under  the  arcs,  and  then 
the  area  of  the  rectilinear  space  between  the  chords  of  the 
arcs  and  the  other  right  lines. 

Thus,  the  Gothic  arch  ACB, 
contains  the  two  segments 
ACH,  BCD,  and  the  plane  tri- 
angle ABC. 

Ex.  If  AB  be  110,  each  of 
the  lines  AC  and  BC  100,  and 
the  height  of  each  of  the  seg- 
ments ACH,  BCD  10.435 ; 
what  is  the  area  of  the  whole  figure  ? 

The  areas  of  the  two  segments  are 
The  area  of  the  triangle  ABC  is 
And  the  whole  figure  is 


82  MENSURATION    OF   THE    CIRCLE. 

PROBLEM  VII. 
To  find  the  area  of  a  circular  ZONE. 

37.  FROM    THE  AREA    OF    THE  WHOLE    CIRCLE,  SUBTRACT 

THE    TWO    SEGMENTS.  ON    THE    SIDES    OF    THE    ZONE. 

If  from  the  whole  circle  there  be  taken  the  two  segments 
ABC  and  DBH,  there  will  remain 
the  zone  ACDH. 

Or,  the  area  of  the  zone  may  be 
found  by  subtracting  the  segment 
ABC  from  the  segment  HBD  :  Or, 
by  adding  the  two  small  segments 
GAH  and  VDC,  to  the  trapezoid 
ACDH.  (See  Art.  36.) 

The  latter  method  is  rather  the 

most  expeditious  in  practice,  as  the  two  segments  at  the  end 
of  the  zone  are  equal. 

Ex.  1.  What  is  the  area  of  the  zone  ACDH,  if  AC  is 
7.75,  DH  6.93,  and  the  diameter  of  the  circle  8  ? 

The  area  of  the  whole  circle  is  50.26 

of  the  segment  ABC  17.32 

of  the  segment  DFH  9.82 

of  the  zone  ACDH  23.12 

2.  What  is  the  area  of  a  zone,  one  side  of  which  is  23.25, 
and  the  other  side  20.8,  in  a  circle  whose  diameter  is  24  ? 

Ans.  208. 

38.  If  the  diameter  of  the  circle  is  not  given,  it  may  be 
found  from  the  sides  and  the  breadth  of  the  zone. 

Let  the  centre  of  the  circle  be  at  O.  Draw  ON  perpen- 
dicular to  AH,  NM  perpendicular  to  LB,  and  HP  perpen- 
dicular to  AL.  Then, 


MENSURATION    OF   THE    CIRCLE.  88 

AN=*AH,  (Euc.  3.  3.)*         MN=i(LA+RH) 
LM=iLR,  (Euc.  2.  6.)f         PA  =LA— RH. 

The  triangles  APH  and  OMN  are  similar,  because  the 
sides  of  one  are  perpendicular  to  those  of  the  other,  each  to 
each,  Therefore, 

PH  :  PA  : :  MN  :  MO 
MO  being  found,  we  have  ML — MO=OL. 


And  the  radius  CO=vOL2+CLa.  (Euc.  47.  l.)J 

Ex.  If  the  breadth  of  the  zone  ACDH  (Fig.  12.)  be  6.4, 
and  the  sides  6.8  and  6 ;  what  is  the  radius  of  the  circle  ? 

PA=3.4— 3=0.4.         And,  MN=i(3.4+3)=3.2. 
Then,  6.4  :  0.4  ::  3.2  :  0.2=MO.     And,  3.2 — 0.2=3=OL 


And  the  radius  CO=V32  +  (3.4)2=4.534. 

PROBLEM  VIII. 
To  find  the  area  of  a  LUNE  or  crescent. 

39.  FIND  THE  DIFFERENCE  OF  THE  TWO  SEGMENTS  WHICH 
ARE  BETWEEN  THE  ARCS  OF  THE  CRESCENT  AND  ITS  CHORD. 

If  the  segment  ABC,  be 
taken  from  the  segment  ABD  ; 
there  will  remain  the  lune  or 
crescent  ACBD. 


Ex.  If  the  chord  AB  be  88, 
the  height  CH  20,  and  the 
height  DH  40 ;  what  is  the 
area  of  the  crescent  ACBD  ? 

The  area  of  the  segment  ABD  is  2698 

of  the  segment  ABC  1220 

of  the  crescent  ACBD  1478 


*  Thomson's  Legendre,  6.  2.         f  Ibid.  15.  4.        $  Ibid.  11.  4. 

2* 


34  MENSURATION    OF   THE    CIRCLE. 


PROBLEM  IX. 

To  find  the  area  of  a  RING,  included  between  the  periphe- 
ries of  two  concentric  circles. 

40.      FlND    THE     DIFFERENCE     OF    THE    AREAS    OF    THE    TWO 
CIRCLES. 

Or, 

Multiply  the  product  of  the  sum  and  difference  of  the  two 
diameters  by  .7854. 

The  area  of  the  ring  is  evidently  equal  to  the  difference 
between  the   areas   of  the   two  cir-  \     B 

cles  AB  and  A'B'. 

But  the  area  of  each  circle  is  equal 
to  the  square  of  its  diameter  multi- 
plied into  .7854.  (Art.  30.)  And 
the  difference  of  these  squares  is  equal 
to  the  product  of  the  sum  and  dif- 
ference of  the  diameters.  (Alg.  191.)  Therefore  the  area  of 
the  ring  is  equal  to  the  product  of  the  sum  and  difference 
of  the  two  diameters  multiplied  by  .7854. 

Ex.  1.  If  AB  be  221,  and  A'B'  106,  what  is  the  area  of 


the  ring?      Ans.  (221aX.7854>- <1069X.7854)=29535. 

2.  If  the  diameters  of  Saturn's  larger  ring  be  205,000 
and  190,000  miles,  how  many  square  miles  are  there  on  one 
side  of  the  ring  ? 

Ans.  395000X15000X-7854=4,653,495,000. 


PROMISCUOUS    EXAMPLES    OF    AREAS. 

Ex.  1.  What  is  the  expense  of  paving  a  street  20  rods 
long  and  2  rods  wide,  at  5  cents  for  a  square  foot  ? 

Ans.  54  4£  dollars. 


MENSURATION    OF   THE    CIRCLE.  36 

2.  If  an  equilateral  triangle  contains  as  many  square  feet 
as  there  are  inches  in  one  of  its  sides  ;  what  is  the  area  of 
the  triangle  ? 

Let  x=ihe  number  of  square  feet  in  the  area. 

/j* 

Then—  -=  the  number  of  linear  feet  in  one  of  the  sides. 
1  "2 

And,  (Art.  11.)  x=i  XVB=- 


eHt* 

Reducing  the  equation,  x=  —  =332.55  the  area. 
V3 

3.  What  is  the  side  of  a  square  whose  area  is  equal  to  that 
of  a  circle  452  feet  in  diameter  ? 


Ans.  V(452)aX- 7854=400.574.  (Arts.  30  and  7.) 

4.  What  is  the  diameter  of  a  circle  which  is  equal  to  a 
square  whose  side  is  36  feet  ? 

Ans.  V(36)a-r 0.7854= 40.62 17.  (Arts.  4  and  32.) 

5.  What  is  the  area  of  a  square  inscribed  in  a  circle  whose 
diameter  is  132  feet? 

Ans.  8712  square  feet.  (Art.  33.) 

6.  How  much  carpeting,  a  yard  wide,  will  be  necessary  to 
cover  the  floor  of  a  room  which  is  a  regular  octagon,  the 
sides  being  eight  feet  each  ?  Ans.  34-^  yards. 

7.  If  the   diagonal  of  a  square  be  16  feet,  what  is  the 
area?  Ans.  128  feet.  (Art.  14.) 

8.  If  a  carriage-wheel  four  feet  in  diameter  revolve  300 
times,  in  going  round  a  circular  green  ;  what  is  the  area  of 
the  green  ? 

Ans.  4154^  sq.  rods,  or  25  acres,  3  qrs.  and  34-fc  rods. 

9.  What  will  be  the  expense  of  papering  the  sides  of  a 
room,  at  10  cents  a  square  yard  ;  if  the  room  be  21  feet  long, 


MENSURATION    OF   THE    CIRCLE. 


18  feet  broad,  and  12  feet  high ;  and  if  there  be  deducted  3 
windows,  each  5  feet  by  3,  two  doors  8  feet  by  4£,  and  one 
fire-place  6  feet  by  4£  ?  Ans.  8  dollars  80  cents. 

10.  If  a  circular  pond  of  water  10  rods  in  diameter  be 
surrounded  by  a  gravelled  walk  8-J-  feet  wide ;  what  is  the 
area  of  the  walk?  Ans.  16£  sq.  rods.  (Art.  40.) 


11.  If  CD,  the  base  of  the 
isosceles  triangle  VCD,  be  60 
feet,  and  the  area  1200  feet; 
and  if  there  be  cut  off,  by  the 
line  LG  parallel  to  CD,  the  tri- 
angle VLG,  whose  area  is  432 
feet ;  what  are  the  sides  of  the 
latter  triangle  ? 

Ans.  30,  30,  and  36  feet. 


12.  What  is  the  area  of  an  equilateral  triangle  inscribed 
in  a  circle  whose  diameter  is  52  feet  ? 

Ans.  878.15  sq.  ft. 

13.  If  a  circular  piece  of  land  is  inclosed  by  a  fence,  in 
which  10  rails  make  a  rod  in  length ;  and  if  the  field  con- 
tains as  many  square  rods,  as  there  are  rails  in  the  fence ; 
what  is  the  value  of  the  land  at  120  dollars  an  acre  ? 

Ans.  942.48  dollars. 

14.  If  the  area  of  the  equilat- 
eral triangle  ABD  be  219.5375 
feet ;  what  is  the  area  of  the  cir- 
cle OBDA,  in  which  the  triangle 
is  inscribed  ? 

The  sides  of  the  triangle  are  each 
22.5167.   (Art.  11.) 

And  the  area  of  the  circle  is  530.93. 


MENSURATION    OF    SOLIDS.  87 

15.  If  6  concentric  circles  are  so  drawn,  that  the  space 
between  the  least  or  1st,  and  the  2d   is  21.2058, 

between  the  2d    and  the  3d   is  35.343, 

between  the  3d    and  the  4th  is    -         49.4802, 
between  the  4th   and  the  5th  is  63.6174, 

between  the  5th  and  the  6th  is  77.7546 ; 

what  are  the  several  diameters,  supposing  the  longest  to  be 

equal  to  6  times  the  shortest  ? 

Ans.  3,  6,  9,  12,  15,  and  18. 

16.  If  the  area  between  two  concentric  circles  be  1202.64 
square  inches,  and  the  diameter  of  the  lesser  circle  be  19 
inches,  what  is  the  diameter  of  the  other  ? 

17.  What  is  the  area  of  a  circular  segment,  whose  height 
is  9,  and  base  24  ? 


SECTION  III. 

SOLIDS    BOUNDED    BY    PLANE    SURFACES. 

ART.  41.  DEFINITION  I.  A  prism  is  a  solid  bounded  by 
plane  figures  or  faces,  two  of  which  are  parallel,  similar,  and 
equal ;  and  the  others  are  parallelograms. 

II.  The  parallel  planes  are  sometimes  called  the  bases  or 
ends;  and  the  other  figures  the  sides  of  the  prism.     The 
latter  taken  together  constitute  the  lateral  surface. 

III.  A  prism  is  right  or  oblique,  according  as  the  sides  are 
perpendicular  or  oblique  to  the  bases. 

IV.  The  height  of  a  prism  is  the  perpendicular  distance 
between  the  planes  of  the  bases.     In  a  right  prism,  there- 
fore, the  height  is  equal  to  the  length  of  one  of  the  sides. 

V.  A  Parallelepiped  is  a  prism  whose  bases  are  parallelo- 
grams. 


88  MENSURATION    OF    SOLIDS. 

VI.  A  Cube  is  a  solid  bounded  by  six  equal  squares.     It 
is  a  right  prism  whose  sides  and  bases  are  all  equal. 

VII.  A  Pyramid  is  a  solid   bounded  by  a  plane  figure 
called  the  base,  and  several  triangular  planes,  proceeding  from 
the  sides  of  the  base,  and  all  terminating  in  a  single  point. 
These  triangles  taken  together  constitute  the  lateral  surface. 

VIII.  A  pyramid  is  regular,  if  its  base  is  a  regular  poly- 
gon, and  if  a  line  from  the  centre  of  the  base  to  the  vertex 
of  the  pyramid  is  perpendicular ..  to  the  base.     This  line  is 
called  the  axis  of  the  pyramid. 

IX.  The  height  of  a  pyramid  is  the  perpendicular  distance 
from  the  summit  to  the  plane  of  the  base.    In  a  regular  pyr- 
amid, it  is  the  length  of  the  axis. 

X.  The  slant-height  of  a  regular  pyramid,  is  the  distance 
from  the  summit  to  the  middle  of  one  of  the  sides  of  the  base. 

XI.  A  frustum  or  trunk  of  a  pyramid  is  a  portion  of  the 
solid  next  the  base,  cut  off  by  a  plane  parallel  to  the  base. 
The   height   of   the   frustum  is  the  perpendicular  distance 
of  the  two  parallel    planes.      The   slant  height  of    a  frus- 
tum of  a  regular  pyramid,  is  the  distance  from  the  middle  of 
one  of  the  sides  of  the  base,  to  the  middle  of  the  corres- 
ponding side  in  the  plane  above.     It  is  a  line  passing  on 
the  surface  of  the  frustum,  through  the  middle  of  one  of 
its  sides. 

XII.  A  Wedge  is  a  solid  of  five  sides,  viz.  a  rectangular 
base,  two  rhomboidal 

sides  meeting  in  an  J& H 

edge,  and  two  tri- 
angular ends ;  as 
ABHG.  The  base 
is  ABCD,  the  sides 
are  ABHG  and 
DCHG,  meeting  in 
the  edge  GH,  and 
the  ends  are  BCH  and  ADG.  The  height  of  the  wedge  is  a 


MENSURATION    OF   SOLIDS.  39 

perpendicular  drawn  from  any  point  in  the  edge,  to  the  plane 
of  the  base,  as  GP. 

XIII.  A  Prismoid  is  a  solid  whose  ends  or  bases  are  par- 
allel, but  not  similar,  and  whose  sides  are  quadrilateral.     It 
differs  from  a  prism  or  a  frustum  of  a  pyramid,  in  having  its 
ends  dissimilar.     It  is  a  rectangular  prismoid,  when  its  ends 
are  right  parallelograms. 

XIV.  A  linear  side  or  edge  of  a  solid  is  the  line  of  intersec- 
tion of  two  of  the  planes  which  form  the  surface. 

42.  The  common  measuring  unit  of  solids  is  a  cube,  whose 
sides  are  squares  of  the  same  name.     The  sides  of  a  cubic 
inch   are  square  inches  ;  of  a  cubic  foot,  square  feet,  <kc. 
Finding  the  capacity,  solidity*  or  solid  contents  of  a  body,  is 
finding  the  number  of  cubic  measures,  of  some  given  de- 
nomination contained  in  the  body. 

In  solid  measure. 

1728     cubic  inches  =1  cubic  foot, 
27     cubic  feet       =1  cubic  yard, 
4492i  cubic  feet       =1  cubic  rod, 
32768000     cubic  rods      =1  cubic  mile, 
282     cubic  inches  =1  ale  gallon, 
231     cubic  inches  =1  wine  gallon, 
2150.42     cubic  inches  =1  bushel, 

1     cubic  foot  of   pure  water   weighs  1000 
avoirdupois  ounces,  or  62£  pounds. 

PROBLEM  I. 
To  find  the  SOLIDITY   of  a  PRISM. 

43.  MULTIPLY  THE  AREA  OF  THE  BASE  BY  THE  HEIGHT. 

This   is   a   general    rule,    applicable    to    parallelepipeds 
whether  right  or  oblique,  cubes,  triangular  prisms,  &c. 

*  See  note  A. 


40  MENSURATION    OF    SOLIDS. 

As  surfaces  are  measured,  by  comparing  them  with  a  right 
parallelogram  (Art.  3.)  ;  so  solids  are  measured,  by  com- 
paring them  with  a  right  parallelepiped. 

If  ABCD  be  the  base  of  a  right     D  o 

parallelepiped,  as  a  stick  of  timber 
standing  erect,  it  is  evident  that  the 
number  of  cubic  feet  contained  in  one 
foot  of  the  height,  is  equal  to  the 
number  of  square  feet  in  the  area  of 
the  base.  And  if  the  solid  be  of  any 

other  height,  instead  of  one  foot,  the  contents  must  have  the 
same  ratio.  For  parallelepipeds  of  the  same  base  are  to 
each  other  as  their  heights.  (Sup.  Euc.  9.  3.)*  The  solidity 
of  a  right  parallelepiped,  therefore,  is  equal  to  the  product 
of  its  length,  breadth,  and  thickness.  See  Alg.  39*7. 

And  an  oblique  parallelepiped  being  equal  to  a  right  one 
of  the  same  base  and  altitude,  (Sup.  Euc.  7.  3)f  is  equal  to 
the  area  of  the  base  multiplied  into  the  perpendicular  height. 
This  is  true  also  of  prisms,  whatever  be  the  form  of  their 
bases.  (Sup.  Euc.  2.  Cor.  to  8,  3.  Thomson's  Legendre,  12.  7.) 

44.  As  the  sides  of  a  cube  are  all  equal,  the  solidity  is 
found  by  cubing  one  of  its  edges.     On  the  other  hand,  if  the 
solid  contents  be  given,  the  length  of  the  edges  may  be 
found,  by  extracting  the  cube  root. 

45.  When  solid  measure  is  cast  by  Duodecimals,  it  is  to 
be  observed  that  inches  are  not  primes  of  feet,  but  thirds. 
If  the  unit  is  a  cubic  foot,  a  solid  which  is  an  inch  thick  and 
a  foot  square  is  a  prime  ;  a  parallelepiped  a  foot  long,  an 
inch  broad,  and  an  inch  thick  is  a  second,  or  the  twelfth  part 
of  a  prime  ;  and  a  cubic  inch  is  a  third,  or  the  twelfth  part 
of  a  second.    A  linear  inch  is  -^  of  a  foot,  a  square  inch  lif 
of  .  a  foot,  and  a  cubic  inch  -p^Vs-  of  a  foot. 


*  Thomson's  Legendre,  9.  7.  f  Ibid.,  7.  7. 


MENSURATION    OP   SOLIDS.  41 

Ex.  1.  What  are  the  solid  contents  of  a  stick  of  timber 
which  is  31  feet  long,  1  foot  3  inches  broad,  and  9  inches 
thick?  Ans.  29  feet  9",  or  29  feet  108  inches. 

2.  What  is  the  solidity  of  a  wall  which  is  22  feet  long,  12 
feet  high,  and  2  feet  6  inches  thick  ? 

Ans.  660  cubic  feet. 

3.  What  is  the  capacity  of  a  cubical  vessel  which  is  2  feet 
3  inches  deep  ? 

Ans.  11  F.  4'  8"  3'",  or  11  feet  675  inches. 

4.  If  the  base  of  a  prism  be  108  square  inches,  and  the 
height  36  feet,  what  are  the  solid  contents  ? 

Ans.  27  cubic  feet. 

5.  If  the  height  of  a  square  prism  be  2^  feet,  and  each 
side  of  the  base  10-^  feet,  what  is  the  solidity  ? 

The  area  of  the  base  =   10iXlOi=106|-  sq.  feet. 
And  the  solid  contents  =106£x    2-^=240^  cubic  feet. 

6.  If  the  height  of  a  prism  be  23  feet,  and  its  base  a  reg- 
ular pentagon,  whose  perimeter  is  18  feet,  what  is  the  so- 
lidity ?  Ans.  512.84  cubic  feet. 

46.  The  number  of  gallons  or  bushels  which  a  vessel  will 
contain  may  be  found,  by  calculating  the  capacity  in  inches, 
and  then  dividing  by  the  number  of  inches  in  1  gallon  or 
bushel. 

The  weight  of  water  in  a  vessel  of  given  dimensions  is 
easily  calculated ;  as  it  is  found  by  experiment,  that  a  cubic 
foot  of  pure  water  weighs  1000  ounces  avoirdupois.  For 
the  weight  in  ounces,  then,  multiply  the  cubic  feet  by  1000 ; 
or  for  the  weight  in  pounds,  multiply  by  62£. 

Ex.  1.  How  many  ale  gallons  are  there  in  a  cistern  which 
is  11  feet  9  inches  deep,  and  whose  base  is  4  feet  2  inches 
square  ? 


MEKSOHATION    OF  SOLIDS. 

The  cistern  contains  352500  cubic  inches; 
And  352500-r282  = 


2.  How  many  wine  gallons  will  fill  a  ditch  3  feet  1 1  inches 
wide,  3  feet  deep,  and  462  feet  long  ?  Ans.  40608. 

3.  What  weight  of  water  can  be  put  into  a  cubical  vessel 
4  feet  deep  ?  Ans.  4000  Ibs. 

PROBLEM  II. 
To  find  the  LATERAL  SURFACE  of  a  RIGHT  PRISM. 

47.  MULTIPLY  THE  LENGTH  INTO  THE  PERIMETER  OF  THE 
BASE. 

Each  of  the  sides  of  the  prism  is  a  right  parallelogram, 
whose  area  is  the  product  of  its  length  and  breadth.  But 
the  breadth  is  one  side  of  the  base ;  and  therefore,  the  sum 
of  the  breadths  is  equal  to  the  perimeter  of  the  base. 

Ex.  1.  If  the  base  of  a  right  prism  be  a  regular  hex- 
agon whose  sides  are  each  2  feet  3  inches,  and  if  the  height 
be  16  feet,  what  is  the  lateral  surface  ? 

Ans.  216  square  feet. 

If  the  areas  of  the  two  ends  be  added  to  the  lateral  sur- 
face, the  sum  will  be  the  whole  surface  of  the  prism.  And 
the  superficies  of  any  solid  bounded  by  planes,  is  evidently 
equal  to  the  areas  of  all  its  sides. 

2.  If  the  base  of  a  prism  be  an  equilateral  triangle 
whose  perimeter  is  6  feet,  and  if  the  height  be  1 7  feet,  what 
is  the  surface  ? 

The  area  of  the  triangle  is     1.732.  {Art.  11.) 

And  the  whole  surface  is  105.464. 


MENSURATION    OP   SOLIDS.  4d 

PROBLEM  III. 

To  find  the  SOLIDITY  of  a  PYRAMID. 
48.  MULTIPLY   THE    AREA   OF  THE  BASE  INTO  •§•  OF  THE 

HEIGHT. 

The  solidity  of  a  prism  is  equal  to  the  product  of  the 
area  of  the  base  into  the  height.  (Art.  43.)  And  a  pyramid 
is  f  of  a  prism  of  the  same  base  and  altitude.  (Sup.  Euc. 
15,  3.  Cor.  1.)*  Therefore  the  solidity  of  a  pyramid 
whether  right  or  oblique,  is  equal  to  the  product  of  the  base 
into  •£  of  the  perpendicular  height. 

Ex.  1.  What  is  the  solidity  of  a  triangular  pyramid, 
whose  height  is  60,  and  each  side  of  whose  base  is  4  ? 

The  area  of  the  base  is      6.928 
And  the  solidity  is          138.56. 

2.  Let  ABC  be  one  side  of  an  oblique 
pyramid  whose  base  is  6  feet  square ; 
let  BC  be  20  feet,  and  make  an  angle 
of  70  degrees  with  the  plane  of  the 
base  ;  and  let  CP  be  perpendicular  to  this 
plane.  What  is  the  solidity  of  the  pyr- 
amid ? 


In  the  right  angled  triangle  BCP,  (Trig.  134.) 
R  I  BC  :  :  sin  B  : :  PC=18.79. 
And  the  solidity  of  the  pyramid  is  225.48  feet, 

3.  What  is  the  solidity  of  a  pyramid  whose  perpendicular 
height  is  72,  and  the  sides  of  whose  base  are  67,  54,  and 
40?  Ans.  25920; 

*  Thomson's  Legendre,  15  and  18.  7. 


44  MENSURATION    OF    SOLIDS. 


PROBLEM  IV. 
To  find  the  LATERAL  SURFACE  of  a  REGULAR  PYRAMID. 

49.  MULTIPLY  HALF  THE  SLANT-HEIGHT  INTO  THE  PERIM- 
ETER OF  THE  BASE. 

Let  the  triangle  ABC  be  one  of 
the  sides  of  a  regular  pyramid.  As 
the  sides  AC  and  BC  are  equal,  the 
angles  A  and  B  are  equal.  Therefore 
a  line  drawn  from  the  vertex  C  to  the 
middle  of  AB  is  perpendicular  to  AB. 
The  area  of  the  triangle  is  equal  to  A 
the  product  of  half  this  perpendicular  into  AB.  (Art.  8.) 
The  perimeter  of  the  base  is  the  sum  of  its  sides,  each  of 
which  is  equal  to  AB.  And  the  areas  of  all  the  equal  tri- 
angles which  constitute  the  lateral  surface  of  the  pyramid, 
are  together  equal  to  the  product  of  the  perimeter  into  half 
the  slant-height  CP. 

The  slant-height  is  the  hypothenuse  of  a  right  angled  tri- 
angle, whose  legs  are  the  axis  of  the  pyramid,  and  the  dis- 
tance from  the  centre  of  the  base  to  the  middle  of  one  of  the 
sides.  See  Def.  10. 

Ex.  1.  What  is  the  lateral  surface  of  a  regular  hexagonal 
pyramid,  whose  axis  is  20  feet,  and  the  sides  of  whose  base 
are  each  8  feet  ? 

The  square  of  the  distance  from  the  centre  of  the  base  to 
one  of  the  sides.  (Art.  16.)=48. 

The  slant-height  (Euc.  47.  l.)*=V4&H-(20)9=21.16 

And  the  lateral  surface=21. 16X4X6  =  507. 84  sq.  feet. 

2.  What  is  the  whole  surface  of  a  regular  triangular  pyr« 
*  Thomson's  Legendre,  11.4. 


MENSURATION    OF    SOLIDS. 


45 


amid  whose  axis  is  8,  and  the  sides  of  whose  base  are  each 

20.78  ? 

The  lateral  surface  is  312 

The  area  of  the  base  is  187 

And  the  whole  surface  is  499 

3.  What  is  the  lateral  surface  of  a  regular  pyramid 
whose  axis  is  12  feet,  and  whose  base  is  18  feet  square  ? 

Ans.  540  square  feet. 

The  lateral  surface  of  an  oblique  pyramid  may  be  found, 
by  taking  the  sum  of  the  areas  of  the  unequal  triangles 
which  form  its  sides. 


PROBLEM  V. 
To  find  the  SOLIDITY  of  a  FRUSTUM  of  a  pyramid. 

50.  ADD  TOGETHER  THE  AREAS  OF  THE  TWO  ENDS,  AND 
THE  SQUARE  ROOT  OF  THE  PRODUCT  OF  THESE  AREAS  J  AND 
MULTIPLY  THE  SUM  BY  \  OF  THE  PERPENDICULAR  HEIGHT 
OF  THE  SOLID, 

Let  CDGL  be  a  vertical 
section,  through  the  middle 
of  a  frustum  of  a  right  pyr- 
amid CDV,  whose  base  is  a 
square. 

Let  CD=a,  LG=&,  RN=A. 

By  similar  triangles, 
LG  :  CD  : :  RV  :  NV. 

Subtracting  the  antecedents,  (Alg.  349.) 
LG  :  CD— LG  : :  RV  :  NV— RV=RN. 

™     f      v       RNxLG      hb 
Therefore  RV=^-^=  — 


46  MENSURATION    OF   SOLIDS. 

The  square  of  CD  is  the  base  v 

of  the  pyramid  CD  V  ; 

And  the  square  of  LG  is 
the  base  of  the  small  pyr- 
amid LGV. 

Therefore,  the   solidity  of  / 

the  larger  pyramid  (Art.  « 

48)  is  c 


—  \  =  _*?! 

a  —  67      Ba  — 


36 
And  the  solidity  of  the  smaller  pyramid  is  equal  to 

LG^XlRV=6'X  — ~         W 
3a — 36~~3a — 36 

If  the  smaller  pyramid   be  taken  from  the  larger,  there 
will  remain  the  frustum.  CDLG,  whose  solidity  is  equal  to 

(Alg.  194.  a.) 
Or,  because  Va262=a6,  (Alg.  210.  a.) 


Here  h,  the  height  of  the  frustum,  is  multiplied  into  a 
and  6*,  the  areas  of  the  two  ends,  and  into  V«262  the  square 
root  of  the  products  of  these  areas. 

In  this  demonstration  the  pyramid  is  supposed  to  be 
square.  But  the  rule  is  equally  applicable  to  a  pyramid  of 
any  other  form.  For  the  solid  contents  of  pyramids  are 
equal,  when  they  have  equal  heights  and  bases,  whatever  be 
the  figure  of  their  bases.  (Sup.  Euc.  14.  3.)*  And  the  sec- 

*  Thomson's  Legendre,  14.  7. 


MENSURATION    OF   SOLIDS.  47 

tions  parallel  to  the  bases,  and  at  equal  distances,  are  equal 
to  one  another.  (Sup.  Euc.  12.  3.  Cor.  2.)* 

Ex.  1.  If  one  end  of  the  frustum  of  a  pyramid  be  9  feet 
square,  the  other  end  6  feet  square,  and  the  height  36  feet, 
what  is  the  solidity  ? 

The  areas  of  the  two  ends  are  81  and  36. 
The  square  root  of  their  product  is  54. 
And  the  solidity  of  the  frustum=(81+36+54)xl2=2052. 

2.  If  the  height  of  a  frustum  of  a  pyramid  be  24,  and 
the  areas  of  the  two  ends  441  and  121 ;  what  is  the  solid- 
ity ?  Ans.  6344. 

3.  If  the  height  of  a  frustum  of  a  hexagonal  pyramid  be 
48,  each  side  of  one  end  26,  and  each  side  of  the  other  end 
16  ;  what  is  the  solidity  ?  Ans.  56034*, 

PROBLEM  VI. 

To  find  the  LATERAL  SURFACE  of  a  FRUSTUM  of  a  regular 
pyramid. 

51.  MULTIPLY  HALF  THE  SLANT-HEIGHT  BY  THE  SUM  OF 
THE  PERIMETERS  OF  THE  TWO  ENDS. 

Each  side  of  a  frustum  of  a  regular  pyramid  is  a  trapezoid, 
as  ABCD.  The  slant-height  HP,  (Def. 
11.)  though  it  is  oblique  to  the  base  of 
the  solid,  is  perpendicular  to  the  line  AB. 
The  area  of  the  trapezoid  is  equal  to  the 
product  of  half  this  perpendicular  into 
the  sum  of  the  parallel  sides  AB  and  DC. 
(Art.  12.)  Therefore  the  area  of  all  the 
equal  trapezoids  which  form  the  lateral 
surface  of  the  frustum,  is  equal  to  the 

*  Thomson's  Legendre,  13,  7.  Cor. 


48 


MENSURATION    OF    SOLIDS. 


product  of  half  the   slant-height  into  the  sum  of  the  pen- 
meters  of  the  ends. 

Ex.  If  the  slant-height  of  a  frustum  of  a  regular  octag- 
onal pyramid  be  42  feet,  the  sides  of  one  end  5  feet  each,  and 
the  sides  of  the  other  end  3  feet  each ;  what  is  the  lateral 
surface  ?  Ans.  1344  square  feet. 

52.  If  the  slant-height  be  not  given,  it  may  be  obtained 
from  the  perpendicular 


height  and  the  dimensions 
of  the  two  ends.     Let  GD 
be  the  slant-height  of   the 
frustum  CDGL,  RN  or  GP               L/ 

V 

\ 

\ 

17 

G 

the    perpendicular    height,              f 
ND  and  RG  the  radii  of  the           / 
circles  inscribed  in  the  pe-        / 

\ 

rimeters  of  the  two  ends.       c 
Then,  PD  is  the  difference  of  the  two  radii 

i 

P          D 

And  the  slant-height  GD=V(GPa+PDa). 


Ex.  If  the  perpendicular  height 
of  a  frustum  of  a  regular  hexagonal 
pyramid  be  24,  the  sides  of  one  end 
13  each,  and  the  sides  of  the  other 
end  8  each ;  what  is  the  whole  sur- 
face? 


V(BC9— BPa)=CP,  that  is,  V(13a— 6.53)=11.258 
And     V88— 4a         =  6.928 
The  difference  of  the  two  radii  is,  therefore        4.33 

The  slant-height=V(24a+4.33a)=24.3875. 
The  lateral  surface  is  1536.4 

And  the  whole  surface,  2 1 4 1 .  Y5 . 


MENSURATION    OF    SOLIDS.  49 

The  height  of  the  whole  pyramid  may  be  calculated  from 
the  dimensions  of  the  frustum.  Let  VN  (Fig.  17.)  be  the 
height  of  the  pyramid,  RN  or  GP  the  height  of  the  frus- 
tum, ND  and  RG  the  radii  of  the  circles  inscribed  in  the 
perimeters  of  the  ends  of  the  frustum. 

Then,  in  the  similar  triangles  GPD  and  VND, 
DP  :  GP :  :  DN  :  VK 

The  height  of  the  frustum  subtracted  from  VN,  gives  VR 
the  height  of  the  small  pyramid  VLG.  The  solidity  and 
lateral  surface  of  the  frustum  may  then  be  found,  by  sub- 
tracting from  the  whole  pyramid,  the  part  which  is  above 
the  cutting  plane.  This  method  may  serve  to  verify  the  cal- 
culations which  are  made  by  the  rules  in  Arts.  50  and  51. 

Ex.  If  one  end  of  the  frustum  CDGL  (Fig.  17.)  be  90  feet 
square,  the  other  end  60  feet  square,  and  the  height  RN  36 
feet ;  what  is  the  height  of  the  whole  pyramid  VCD  :  and 
what  are  the  solidity  and  lateral  surface  of  the  frustum  ? 

DP=DN—GR=45— 30=15.       And,  GP=RN=36. 

Then,  15  :  36  : :  45  :  108=VN,  the  height  of  the  whole 
pyramid. 

And,  108— 36=72=VR,  the  height  of  the  part  VLG." 

The  solidity  of  the  large  pyramid  is  291600  (Art.  48.) 
of  the  small  pyramid        86400 

of  the  frustum  CDGL    205200 


The  lateral  surface  of  the  large  pyramid  is    21060  (Art.  49.) 
of  the  small  pyramid         9360 

of  the  frustum  11700 


MENSURATION    OF    SOLIDS. 


PROBLEM  VII. 
To  find  the  SOLIDITY  of  a  WEDGE. 

54.  ADD  THE  LENGTH  OF  THE  EDGE  TO  TWICE  THE 
LENGTH  OF  THE  BASE,  AND  MULTIPLY  THE  SUM  BY  £  OF  THE 

PRODUCT  OF  THE  HEIGHT  OF  THE  WEDGE  AND  THE  BREADTH 
OF  THE  BASE. 

Let    L  =  AB    the 

length  of  the  base. 
Let  Z=GH  the  length 

of  the  edge. 
Let  6=BC  the  breadth 

of  the  base. 
Let  A=PG  the  height 

of  the  wedge. 

Then,  L—  /=AB—  GH=AM. 

If  the  length  of  the  base  and  the  edge  be  equal,  as  BM 
and  GH,  the  wedge  MBHG  is  half  a  parallelepiped  of  the 
same  base  and  height.  And  the  solidity  (Art.  43.)  is  equal 
to  half  the  product  of  the  height,  into  the  length  and  breadth 
of  the  base  ;  that  is  \  Ihl. 

If  the  length  of  the  base  be  greater  than  that  of  the  edge, 
as  -ABGH  ;  let  a  section  be  made  by  the  plane  GMN,  par- 
allel to  HBC.  This  will  divide  the  whole  wedge  into  two 
parts  MBHG  and  AMG.  The  latter  is  a  pyramid,  whose 
solidity  (Art.  48.)  is  i  bhx(L—l) 

The  solidity  of  the  parts  together,  is,  therefore, 


If  the  length  of  the  base  be  less  than  that  of  the  edge,  it 
is  evident  that  the  pyramid  is  to  be  subtracted  from  half  the 
parallelepiped,  which  is  equal  in  height  and  breadth  to  the 
wedge,  and  equal  in  length  to  the  edge. 


MENSURATION    OF    SOLIDS. 

The  solidity  of  the  wedge  is,  therefore, 


51 


Ex.  1.  If  the  base  of  a  wedge  be  35  by  15,  the  edge  55, 
and  the  perpendicular  height  12.4  ;  what  is  the  solidity? 

Ans. 


2.  If  the  base  of  a  wedge  be  27  by  8,  the  edge  36,  and 
the  perpendicular  height  42  ;  what  is  the  solidity  ? 

Ans.  5040. 

PROBLEM  VIII. 
To  find  the  SOLIDITY  of  a  rectangular  PRISMOID. 

55.  To  THE  AREAS  OF  THE  TWO  ENDS,  ADD  FOUR  TIMES 
THE  AREA  OF  A  PARALLEL  SECTION  EQUALLY  DISTANT  FROM 
THE  ENDS,  AND  MULTIPLY  THE  SUM  BY  •£  OF  THE  HEIGHT. 

Let   L    and    B    be   the    length   and 

breadth  of  one  end, 
Let  I  and  b  be  the  length  and  breadth 

of  the  other  end, 
Let   M  and    m   be  the    length    and 

breadth  of  the  section  in  the  middle. 
And  h   be  the    height   of   the    pris- 

moid. 

The  solid  may  be  divided  into  two  wedges  whose  bases  are 
the  ends  of  the  prismofd,  and  whose  edges  are  L  and  /.  The 
solidity  of  the  whole,  by  the  preceding  article  is, 


\ 


As  M  is  equally  distant  from  L  and  I, 


2M=L+Z, 


[bL+lb. 


52  MENSURATION  OF  SOLIDS. 

Substituting  4  Mm  for  its  value,  in  the   preceding  ex- 
pression for  the  solidity,  we  have 


That  is,  the  solidity  of  the  prismoid  is  equal  to  •§•  of  the 
height,  multiplied  into  the  areas  of  the  two  ends,  and  4 
times  the  area  of  the  section  in  the  middle. 

This  rule  may  be  applied  to  prismoids  of  other  forms. 
For,  whatever  be  the  figure  of  the  two  ends,  there  may  be 
drawn  in  each,  such  a  number  of  small  rectangles,  that  the 
sum  of  them  shall  differ  less,  than  by  any  given  quantity, 
from  the  figure  in  which  they  are  contained.  And  the  solids 
between  these  rectangles  will  be  rectangular  prismoids. 

Ex.  1.  If  one  end  of  a  rectangular  prismoid  be  44  feet  by 
23,  the  other  end  36  by  21,  and  the  perpendicular  height 
72  ;  what  is  the  solidity  ? 

The  area  of  the  larger  end         =44X23  =  1012 
of  the  smaller  end       =36X21=   756 
of  the  middle  section  =40X22=   880 
And  the  solidity=(1012-l-756-f4X880)x  12  =  63456  feet. 

2.  What  is  the  solidity  of  a  stick  of  hewn  timber,  whose 
ends  are  30  inches  by  27,  and  24  by  18,  and  whose  length 
is  48  feet?  Ans.  204  feet. 

Other  solids  not  treated  of  in  this  section,  if  they  be 
bounded  by  plane  surfaces,  may  be  measured  by  supposing 
them  to  be  divided  into  prisms,  pyramids,  and  wedges.  And, 
indeed,  every  such  solid  may  be  considered  as  made  up  of 
triangular  pyramids. 


MENSURATION    OF    REGULAR   SOLIDS.  53 


THE   FIVE    REGULAR    SOLIDS, 

56.  A  SOLID  IS  SAID  TO  BE  REGULAR,  WHEN  ALL  ITS 
BOLID  ANGLES  ARE  EQUAL,  AND  ALL  ITS  SIDES  ARE  EQUAL 
AND  REGULAR  POLYGONS. 

The  following  figures  are  of  this  description ; 
1.  The  Tetraedron, 


2.  The  Hexaedron  or  cube, 

3.  The  Octaedron, 

4.  The  Dodecaedron, 

5.  The  Icosaedron, 


whose 
sides  are 


four  triangles ; 
six  squares ; 
eight  triangles ; 
twelve  pentagons; 
L  twenty  triangles.* 


Besides  these  five  there  can  be  no  other  regular  solids. 
The  only  plane  figures  which  can  form  such  solids,  are  tri- 
angles, squares,  and  pentagons.  For  the  plane  angles  which 
contain  any  solid  angle,  are  together  less  than  four  right  an- 
gles or  360°.  (Sup.  Euc.  21,  2.)  And  the  least  number 
which  can  form  a  solid  angle  is  three.  (Sup.  Euc.  Def.  8,  2.) 
If  they  are  angles  of  equilateral  triangles,  each  is,  60°.  The 
sum  of  three  of  them  is  180°,  of  four  240°,  of  Jive  300°,  and 
of  six  360°.  The  latter  number  is  too  great  for  a  solid  angle. 

The  angles  of  squares  are  90°  each.  The  sum  of  three  of 
these  is  270°,  of  four  360°,  and  of  any  other  greater  number, 
still  more. 

The  angles  of  regular  pentagons  are  108°  each.  The  sum 
of  three  of  them  is  324°  :  of  four,  or  any  other  greater  num- 
ber, more  than  360°.  The  angles  of  all  other  regular  poly- 
gons are  still  greater. 

In  a  regular  solid,  then,  each  solid  angle  must  be  con- 
tained by  three,  four,  or  five  equilateral  triangles,  by  three 
squares,  or  by  three  regular  pentagons. 

*  For  the  geometrical  construction  of  these  solids,  see  Legendre's 
Geometry ;  Appendix  to  Books  VI.  and  VII.,  or  Thomson's  Legendre, 
p.  214. 


54  MENSURATION    OF   REGULAR    SOLIDS. 

57.  As  the  sides  of  a  regular  solid  are  similar  and  equal, 
and  the  angles  are  also  alike  ;  it  is  evident  that  the  sides 
are  all  equally  distant  from  a  central  point  in  the  solid.  If 
then,  planes  be  supposed  to  proceed  from  the  several  edges 
to  the  centre,  they  will  divide  the  solid  into  as  many  equal 
pyramids,  as  it  has  sides.  The  base  of  each  pyramid  will  be 
one  of  the  sides ;  their  common  vertex  will  be  the  central 
point;  and  their  height  will  be  a  perpendicular  from  the 
centre  to  one  of  the  sides. 


PROBLEM  IX. 
To  find  the  SURFACE  of  a  REGULAR  SOLID. 

58.  MULTIPLY  THE  AREA  OF  ONE  OF  THE  SIDES  BY  THE 
NUMBER  OF  SIDES. 

Or, 

MULTIPLY  THE  SQUARE  OF  ONE  OF  THE  EDGES,  BY  THE 
SURFACE  OF  A  SIMILAR  SOLID  WHOSE  EDGES  ARE  1. 

As  all  the  sides  are  equal,  it  is  evident  that  the  area  of 
one  of  them,  multiplied  by  the  number  of  sides,  will  give  the 
area  of  the  whole. 

Or,  if  a  table  is  prepared,  containing  the  surfaces  of  the 
several  regular  solids  whose  linear  edges  are  unity  ;  this  may 
be  used  for  other  regular  solids,  upon  the  principle,  that  the 
areas  of  similar  polygons  are  as  the  squares  of  their  homolo- 
gous sides.  (Euc.  20.  6.)*  Such  a  table  is  easily  formed,  by 
multiplying  the  area  of  one  of  the  sides,  as  given  in  Art.  1 7, 
by  the  number  of  sides.  Thus,  the  area  of  an  equilateral 
triangle  whose  side  is  1,  is  0.4330127.  Therefore,  the  sur- 
face 

*  Thomson's  Legendre,  27.  4. 


MENSURATION    OF   REGULAR   SOLIDS.  55 

Of  a  regular  tetraedron  =.4330127x4  =1.7320508. 
Of  a  regular  octaedron  =.4330127X8  =3.4641016. 
Of  a  regular  icosaedron  =.4330127X20=8.6602540. 

See  the  table  in  the  following  article. 

Ex.  1.  What  is  the  surface  of  a  regular  dodecaedron  whose 
edges  are  each  25  inches  ? 

The  area  of  one  of  the  sides  is  1075.3 
And  the  surface  of  the  whole  solid  =1075.3X12=12903.6. 

2.  What  is  the  surface  of  a  regular  icosaedron  whose 
edges  are  each  102?  Ans.  90101.3. 

PROBLEM  X. 

To  find  the  SOLIDITY  of  a  REGULAR  SOLID. 
59.  MULTIPLY  THE  SURFACE  BY  -J-  OF  THE  PERPENDICULAB 

DISTANCE    FROM   THE    CENTRE   TO    ONE    OF   THE   SIDES. 

Or, 

MULTIPLY  THE  CUBE  OF  ONE  OF  THE  EDGES,  BY  THE 
SOLIDITY  OF  A  SIMILAR  SOLID  WHOSE  EDGES  ARE  1. 

As  the  solid  is  made  up  of  a  number  of  equal  pyramids, 
whose  bases  are  the  sides,  and  whose  height  is  the  perpendic- 
ular distance  of  the  sides  from  the  centre  (Art.  57.) ;  the 
solidity  of  the  whole  must  be  equal  to  the  areas  of  all  the 
sides  multiplied  into  -£  of  this  perpendicular.  (Art.  48.) 

If  the  contents  of  the  several  regular  solids  whose  edges 
are  1,  be  inserted  in  a  table,  this  may  be  used  to  measure 
other  similar  solids.  For  two  similar  regular  solids  contain 
the  same  number  of  similar  pyramids ;  and  these  are  to  each 
other  as  the  cubes  of  their  linear  sides  or  edges.  (Sup.  Euc. 
15.  3.  Cor.  3.)* 

•  Thomson's  Legendre,  20.  7. 


MENSURATION    OS1    THE    CYLINDER. 


A    TABLE    OF    REGULAR    SOLIDS    WHOSE    EDGES    ARE    1. 


Names.                   JNo.  of  sides.)          Surfaces. 

Solidities. 

Tetraedron 
Hexaedron 
Octaedron 
Dodecaedron 
Icosaedron 

4 
6 

8 
12 
20 

1.7320508 
6.0000000 
3.4641016 

20.6457288 
8.6602540 

0.1178513 
1.0000000 
0.4714045 
7.6631189 
2.1816950 

For  the  method  of  calculating  the  last  column  of  this  table, 
see  Button's  Mensuration,  Part.  III.  Sec.  2. 

Ex.  What  is  the  solidity  of  a  regular  octaedron  whose 
edges  are  each  32  inches  ?  Ans.  15447  inches. 


SECTION  IV. 

THE    CYLINDER,    CONE,    AND    SPHERE. 

ART.  61.  DEFINITION  I.  A  right  cylinder  is  a  solid  de- 
scribed by  the  revolution  of  a  rectangle  about  one  of  its 
sides.  The  ends  or  bases  are  evidently  equal  and  parallel 
circles.  And  the  axis,  which  is  a  line  passing  through  the 
middle  of  the  cylinder,  is  perpendicular  to  the  bases. 

The  ends  of  an  oblique  cylinder  are  also  equal  and  paral- 
lej  circles  ;  but  they  are  not  perpendicular  to  the  axis.  The 
height  of  a  cylinder  is  the  perpendicular  distance  from  one 
base  to  the  plane  of  the  other.  In  a  right  cylinder,  it  is  the 
length  of  the  axis. 

II.  A  right  cone  is  a  solid  described  by  the  revolution  of 
a  right  angled  triangle  about  one  of  the  sides  which  contain 
the  right  angle.  The  base  is  a  circle,  and  is  perpendicular  to 


MENSURATION    OF   THE    CYLINDER. 


the  axis,  which  proceeds  from  the  middle  of  the  base  to  the 
vertex. 

The  base  of  an  oblique  cone  is  also  a  circle,  but  is  not  per- 
pendicular to  the  axis.  The  height  of  a  cone  is  the  perpen- 
dicular distance  from  the  vertex  to  the  plane  of  the  base.  In 
a  right  cone,  it  is  the  length  of  the  axis.  The  slant-height 
of  a  right  cone  is  the  distance  from  the  vertex  to  the  circum- 
ference of  the  base. 

III.  A.  frustum  of  a  cone  is  a  portion  cut  off  by  a  plane 
parallel  to  the  base.     The  height  of  the 

frustum  is  the  perpendicular  distance  of 
the  two  ends.  The  slant-height  of  a 
frustum  of  a  right  cone,  is  the  distance 
between  the  peripheries  of  the  two 
ends,  measured  on  the  outside  of  the 
solid ;  as  AD. 

IV.  A  sphere  or  globe  is  a  solid  which 
has  a  centre  equally  distant  from  every 

part  of  the  surface.  It  may  be  described  by  the  revolution 
of  a  semicircle  about  a  diameter.  A  radius  of  the  sphere  is 
a  line  drawn  from  the  centre  to  any  part  of  the  surface.  A 
diameter  is  a  line  passing  through  the  centre,  and  terminated 
at  both  ends  by  the  surface.  The  circumference  is  the  same 
as  the  circumference  of  a  circle  whose  plane  passes  through 
the  centre  of  the  sphere.  Such  a  circle  is  called  a  great 
circle. 

V.  A  segment  of  a  sphere  is  a  part  cut  off  by  any  plane. 
The  height  of  the  segment  is  a  per- 
pendicular from  the  middle  of   the 

base  to  the  convex  surface,  as  LB. 

VI.  A  spherical  zone  or  frustum  is 
a   part  of  the  sphere  included  be- 
tween two  parallel  planes.      It  is 
called  the  middle  zone,  if  the  planes 
are  equally  distant  from  the  centre. 

3* 


58  MENSURATION    OF   THE    CYLINDER. 

The  height  of  a  zone  is  the  distance  of  the  two  planes,  as 
LR* 

VII.  A  spherical  sector  is  a  solid  produced  by  a  circular 
sector,  revolving  in  the  same  manner 

as  the  semicircle  which  describes  the 
whole  sphere.  Thus  a  spherical  sec- 
tor is  described  by  the  circular  sec- 
tor AGP  or  GCE  revolving  on  the 
axis  CP. 

VIII.  A  solid  described  by   the 
revolution  of  any  figure  about  a  fixed 
axis,  is  called  a  solid  of  revolution. 

PROBLEM  I. 

To  find  the  CONVEX  SURFACE  of  a  RIGHT  CYLINDER. 
62.  MULTIPLY  THE  LENGTH   INTO  THE  CIRCUMFERENCE  OF 

THE    BASE. 

If  a  right  cylinder  be  covered  with  a  thin  substance  like 
paper,  which  can  be  spread  out  into  a  plane  ;  it  is  evident 
that  the  plane  will  be  a  parallelogram,  whose  length  and 
breadth  will  be  equal  to  the  length  and  circumference  of  the 
cylinder.  The  area  must,  therefore,  be  equal  to  the  length 
multiplied  into  the  circumference.  (Art.  4.) 

Ex.  1.  What  is  the  convex  surface  of  a  right  cylinder 
which  is  42  feet  long,  and  15  inches  in  diameter? 

Ans.  42X1-25X3.14159  =  164.933  sq.  feet. 

2.  What  is  the  whole  surface  of  a  right  cylinder,  which 
is  2  feet  in  diameter  and  36  feet  long  ? 

*  According  to  some  writers,  a  spherical  segment  is  either  a  solid 
which  is  cut  off  from  the  sphere  by  a  single  plane,  or  one  which  is  in- 
cluded between  two  planes :  and  a  zone  is  the  surface  of  either  of  these. 
In  this  sense,  the  term  zone  is  commonly  used  in  geography. 


MENSURATION    OF   THE    CYLINDER.  59 

The  convex  surface  is  226.1945 

The  area  of  the  two  ends  (Art.  30.)  is          6.2832 
The  whole  surface  is  232.4777 

3.  What  is  the  whole  surface  of  a  right  cylinder  whose 
axis  is  82,  and  circumference  71  ?  Ans.  6624.32. 

63.  It  will  be  observed  that  the  rules  for  the  prism  and 
pyramid  in  the  preceding  section,  are  substantially  the  same, 
as  the  rules  for  the  cylinder  and  cone  in  this.    There  may  be 
some  advantage,  however,  in  considering  the  latter  by  them- 
selves. 

In  the  base  of  a  cylinder,  there  may  be  inscribed  a  poly- 
gon, which  shall  differ  from  it  less  than  by  any  given  space. 
(Sup.  Euc.  6.  1.  Cor.)*  If  the  polygon  be  the  base  of  a 
prism,  of  the  same  height  as  the  cylinder,  the  two  solids 
may  differ  less  than  by  any  given  quantity.  In  the  same 
manner,  the  base  of  a  pyramid  may  be  a  polygon  of  so  many 
sides,  as  to  differ  less  than  by  any  given  quantity,  from  the 
base  of  a  cone  in  which  it  is  inscribed.  A  cylinder  is  there- 
fore considered,  by  many  writers,  as  a  prism  of  an  infinite 
number  of  sides ;  and  a  cone,  as  a  pyramid  of  an  infinite 
number  of  sides.  (For  the  meaning  of  the  term  "  infinite," 
when  used  in  the  mathematical  sense,  see  Alg.  Sec.  XV.) 

PROBLEM  II. 
To  find  the  SOLIDITY  of  a  CYLINDER. 

64.  MULTIPLY  THE  AREA  OF  THE  BASE  BY  THE  HEIGHT. 

The  solidity  of  a  parallelopiped  is  equal  to  the  product  of 
the  base  into  the  perpendicular  altitude.  (Art.  43.)  And  a 
parallelopiped  and  a  cylinder  which  have  equal  bases  and 
altitudes  are  equal  to  each  other.  (Sup.  Euc.  17.  3.)t 

•  Thomson's  Legendre,  9.  5. 


60 


MENSURATION    OF    THE    CYLINDER. 


Ex.  1.  What  is  the  solidity  of  a  cylinder,  whose  height  is 
121,  and  diameter  45.2  ? 

Ans.  45.2aX. '7854X121  =  194156.6. 

2.  What  is  the  solidity  of  a  cylinder,  whose  height  is  424, 
and  circumference  213  ?  Ans.   1530837. 


3.  If  the  side  AC  of  an  oblique 
cylinder  be  27,  and  the  area  of  the  base 
32.61,  and  if  the  side  make  an  angle 
of  62°  44'  with  the  base,  what  is  the 
solidity  ? 

E  :  AC  : :  sin  A  :  BC=24  the   per- 
pendicular height. 

And  the  solidity  is  782.64. 


4.  The  Winchester  bushel  is  a  hollow  cylinder,  18^- inches 
in  diameter,  and  8  inches  deep.     What  is  its  capacity  ? 
The  area  of  the  base=(18.5)2X- 7853982  =  268.8025. 
And  the  capacity  is  2150.42   cubic  inches.     See  the 
table  in  Art.  42. 


PROBLEM  III. 
To  find  the  CONVEX  SURFACE  of  a  RIGHT  CONE. 

65.  MULTIPLY  HALF  THE  SLANT-HEIGHT  INTO  THE  CIR- 
CUMFERENCE OF  THE  BASE. 

If  the  convex  surface  of  a  right  cone  be  spread  out  into  a 
plane,  it  will  evidently  form  a  sector  of  a  circle  whose  radius 
is  equal  to  the  slant-height  of  the  cone.  But  the  area  of  the 
sector  is  equal  to  the  product  of  half  the  radius  into  the 
length  of  the  arc.  (Art.  34.)  Or  if  the  cone  be  considered 
as  a  pyramid  of  an  infinite  number  of  sides,  its  lateral  sur- 


MENSURATION    OF    THE    CONE.  61 

face  is  equal  to  the  product  of  half  the  slant-height  into  the 
perimeter  of  the  base.  (Art.  49.) 

Ex.  1.  If  the  slant-height  of  a  right  cone  be  82,  and  the 
diameter  of  the  base  24,  what  is  the  convex  surface  ? 

Ans.  41X24X3.14159  =  3091.3  square  feet. 

2.  If  the  axis  of  a  right  cone  be  48,  and  the  diameter  of 
the  base  72,  what  is  the  whole  surface  ? 

The  slant-height  =  V(36Q+48a)=60.  (Euc.  47.  1.) 

The  convex  surface  is  6786 

The  area  of  the  base  4071.6 

And  the  whole  surface  10857.6 

3.  If  the  axis  of  a  right  cone  be  16,  and  the  circumfer- 
ence of  the  base  75.4  ;  what  is  the  whole  surface  ? 

Ans.  1206.4. 

PROBLEM  IV. 
To  find  the  SOLIDITY  of  a  CONE. 

66.  MULTIPLY  THE  AREA  OF  THE  BASE  INTO  •£  OF  THE 
HEIGHT. 

The  solidity  of  a  cylinder  is  equal  to  the  product  of  the 
base  into  the  perpendicular  height.  (Art.  64.)  And  if  a  cone 
and  a  cylinder  have  the  same  base  and  altitude,  the  cone  is 
$  of  the  cylinder.  (Sup.  Euc.  18.  3.)*  Or  if  a  cone  be  con- 
sidered  as  a  pyramid  of  an  infinite  number  of  sides,  the  so- 
lidity is  equal  to  the  product  of  the  base  into  -J  of  the  height, 
by  Art.  48. 

Ex.  1.  What  is  the  solidity  of  a  right  cone  whose  height 
is  663,  and  the  diameter  of  whose  base  is  101  ? 

Ans.  loTaX.7854x221  = 


*  Thomson's  Legendre,  4.  8.  Cor. 


62  MENSURATION    OF    THE    CONE. 

2.  If  the  axis  of  an  oblique  cone  be  738,  and  make  an 
angle  of  30°  with  the  plane  of  the  base ;  and  if  the  circum- 
ference of  the  base  be  355,  what  is  the  solidity  ? 

Ans.  1233536. 

PROBLEM  V. 

To  find  the  CONVEX  SURFACE  of  a  FRUSTUM  of  a  right  cone. 
67.    MULTIPLY  HALF  THE  SLANT-HEIGHT    BY  THE    SUM    OP 

THE    PERIPHERIES    OF    THE    TWO    ENDS. 

This  is  the  rule  for  a  frustum  of  a  pyramid  ;  (Art.  51.) 
and  is  equally  applicable  to  a  frustum  of  a  cone,  if  a  cone  be 
considered  as  a  pyramid  of  an  infinite  number  of  sides. 
(Art.  63.) 

Or  thus, 

Let  the  sector  ABV  represent  the  Ai 
convex  surface  of  a  right  cone,  (Art. 
65.)  and  DCV  the  surface  of  a  portion 
of  the  cone,  cut  off  by  a  plane  parallel 
to  the  base.  Then  will  ABCD  be  the 
surface  of  the  frustum. 

Let  AB=a,  DC =6,  VD=c?,  AD=h.  $ 

Then  the  area  ASV=^ax(h-\-d)=^ah+^ad.  (Art.  34.) 

And  the  area  VCV=±bd. 
Subtracting  the  one  from  the  other, 

The  area  A.EDC=iah+±ad— $bd. 

But  d  :  d+h  ::b  :  a.    (Sup.  Euc.  8.  1.)*     Therefore  iad— 
The  surface  of  the  frustum  then,  is  equal  to 

•••'  •    .  i.  .i-n  .,  —  ... 

*  Thomson's  Legendre,  10.  5.  Cor. 


MENSURATION    OF   THE    CONE.  63 

Cor.  The  surface  of  the  frustum  is  equal  to  the  product 
of  the  slant-height  into  the  circumference  of  a  circle  which 
is  equally  distant  from  the  two  ends.  Thus,  the  surface 
ABCD  is  equal  to  the  product  of  AD  into  MN.  For  MIST  is 
equal  to  half  the  sum  of  AB  and  DC. 

Ex.  1.  What  is  the  convex  surface  of  a  frustum  of  a  right 
cone,  if  the  diameters  of  the  two  ends  be  44  and  33,  and 
the  slant-height  84  ?  Ans.  10159.8. 

2.  If  the  perpendicular  height  of  a  frustum  of  a  right 
cone  be  24,  and  the  diameters  of  the  two  ends  80  and  44, 
what  is  the  whole  surface  ? 

Half  the  difference  of  the  diameters  is  18. 


And  V  18a+24a=30,  the  slant-height,  (Art.  52.) 
The  convex  surface  of  the  frustum  is          5843 
The  sum  of  the  areas  of  the  two  ends  is  6547 

And  the  whole  surface  is  12390 


PROBLEM  VI. 
To  find  the  SOLIDITY  of  a  FRUSTUM  of  a  cone. 

68.  ADD  TOGETHER  THE  AREAS  OF  THE  TWO  ENDS,  AND 
THE  SQUARE  ROOT  OF  THE  PRODUCT  OF  THESE  AREAS  ',  AND 
MULTIPLY  THE  SUM  BY  £  OF  THE  PERPENDICULAR  HEIGHT. 

This  rule,  which  was  given  for  the  frustum  of  a  pyramid, 
(Art.  50.)  is  equally  applicable  to  the  frustum  of  a  cone ;  be- 
cause a  cone  and  a  pyramid  which  have  equal  bases  and  alti- 
tudes are  equal  to  each  other. 

Ex.  1.  What  is  the  solidity  of  a  mast  which  is  72  feet 
long,  2  feet  in  diameter  at  one  end,  and  18  inches  at  the 
other?  Ans.  174.36  cubic  feet. 


64 


MENSURATION    OF   THE    SPHERE. 


2.  What  is  the  capacity  of  a  conical  cistern  which  is  9 
feet  deep,  4  feet  in  diameter  at  the  bottom,  and  3  feet  at  the 
top  ?  Ans.  87.18  cubic  feet^652.15  wine  gallons. 

3.  How  many  gallons  of  ale  can  be  put  into  a  vat  in  the 
form  of  a  conic  frustum,  if  the  larger  diameter  be  7  feet,  the 
smaller  diameter  6  feet,  and  the  depth  8  feet  ? 

PROBLEM  VII. 

To  find  the  SURFACE  of  a  SPHERE. 
69.  MULTIPLY  THE  DIAMETER  BY  THE  CIRCUMFERENCE. 

Let  a  hemisphere  be  described  by  the  quadrant  CPD, 
revolving  on  the  line  CD.  Let 
AB  be  the  side  of  a  regular  poly- 
gon inscribed  in  the  circle  of 
which  DBF  is  an  arc.  Draw  AO 
and  BN  perpendicular  to  CD, 
and  BH  perpendicular  to  AO. 
Extend  AB  till  it  meets  CD  con- 
tinued. The  triangle  AOV,  re- 
volving on  0V  as  an  axis,  will 
describe  a  right  cone.  (Defin.  2.) 
AB  will  be  the  slant-height  of  a 
frustum  of  this  cone  extending 
from  AO  to  BN.  From  G  the  middle  of  AB,  draw  GM 
parallel  to  AO.  The  surface  of  the  frustum  described  by 
AB.  (Art.  67.  Cor.)  is  equal  to 

AKXcirc  GM.* 

From  the  centre  C  draw  CG,  which  will  be  perpendicular 
to  AB,  (Euc.  3.  3.)  and  the  radius  of  a  circle  inscribed  in 


*  By  circ  GM  is  meant  the  circumference  of  a  circle  the  radius  of 
which  is  GM. 


MENSURATION  OF  THE  SPHERE.  60 

the  polygon.     The  triangles  ABH  and  CGM  are  similar,  be- 
cause the  sides  are  perpendicular,  each  to  each.     Therefore, 

HB  or  ON  :  AB  : :  GM  :  GO  : :  circ  GM  :  circ  GO. 

So  that  ON  X  circ  GC=ABx«>c  GM,  that  is,  the  sur- 
face of  the  frustum  is  equal  to  the  product  of  ON  the  per- 
pendicular height,  into  circ  GC,  the  perpendicular  distance 
from  the  centre  of  the  polygon  to  one  of  the  sides. 

In  the  same  manner  it  may  be  proved,  that  the  surfaces 
produced  by  the  revolution  of  the  lines  BD  and  AP  about 
the  axis  DC,  are  equal  to 

NDxcirc  GC,  and  COx^Vc  GC. 

The  surface  of  the  whole  solid,  therefore,  (Euc.  1.2.)  is  equal  to 

GDxcirc  GC. 

The  demonstration  is  applicable  to  a  solid  produced  by 
the  revolution  of  a  polygon  of  any  number  of  sides.  But  a 
polygon  may  be  supposed  which  shall  differ  less  than  by 
any  given  quantity  from  the  circle  in  which  it  is  inscribed  ; 
(Sup.  Euc.  4.  1.)*  and  in  which  the  perpendicular  GC  shall 
differ  less  than  by  any  given  quantity  from  the  radius  of  the 
circle.  Therefore,  the  surface  of  a  hemisphere  is  equal  to 
the  product  of  its  radius  into  the  circumference  of  its  base ; 
and  the  surface  of  a  sphere  is  equal  to  the  product  of  its 
diameter  into  its  circumference. 

Cor.  1.  From  this  demonstration  it  follows,  that  the  sur- 
face of  any  segment  or  zone  of  a  sphere  is  equal  to  the 
product  of  the  height  of  the  segment  or  zone  into  the  cir- 
cumference of  the  sphere.  The  surface  of  the  zone  pro- 
duced by  the  revolution  of  the  arc  AB  about  ON,  is  equal 
to  ON  x  circ  CP.  And  the  surface  of  the  segment  pro- 

*  Thomson's  Legendre,  9.  5. 


MENSURATION    OP    THE    SPHERE. 


duced  by  the  revolution  of  BD  about  DN  is  equal  to 
eirc  CP. 

Cor.  2.  The  surface  of  a  sphere  is  equal  to  four  times  the 
area  of  a  circle  of  the  same  diameter ;  and  therefore,  the 
convex  surface  of  a  hemisphere  is  equal  to  twice  the  area  of 
its  base.  For  the  area  of  a  circle  is  equal  to  the  product  of 
half  the  diameter  into  half  the  circumference  ;  (Art.  30.) 
that  is,  to  •£  the  product  of  the  diameter  and  circumference. 

Cor.  3.  The  surface  of  a  sphere,  or  the  convex  surface 
of  any  spherical  segment  or  zone, 
is  equal  to  that  of  the  circum- 
scribing cylinder.  A  hemis- 
phere described  by  the  revolu- 
tion of  the  arc  DBF,  is  cir- 
cumscribed by  a  cylinder  pro- 
duced by  the  revolution  of  the 
parallelogram  DdCP.  The  con- 
vex surface  of  the  cylinder  is 
equal  to  its  height  multiplied 
by  its  circumference.  (Art.  62.) 
And  this  is  also  the  surface  of 
the  hemisphere. 

So  the  surface  produced  by  the  revolution  of  AB  is  equal 
to  that  produced  by  the  revolution  of  ab.  And  the  surface 
produced  by  BD  is  equal  to  that  produced  by  bd. 

Ex.  1.  Considering  the  earth  as  a  sphere  7930  miles  in 
diameter,  how  many  square  miles  are  there  on  its  surface  ? 

Ans.   197,558,500. 

2.  If  the  circumference  of  the  sun  be  2,800,000,  what  is 
his  surface  ?  Ans.  2,495,547,600,000  sq.  miles. 

3.  How  many  square  feet  of  lead  will  it  require,  to  cover 
a  hemispherical  dome  whose  base  is  13  feet  across  ? 

Ans. 


MENSURATION    OF    THE    SPHERE.  67 

PROBLEM  VIII. 

To  find  the  SOLIDITY  of  a  SPHERE. 
YO.  1.  MULTIPLY  THE  CUBE  OF  THE  DIAMETER  BY  .5230. 

Or, 

2.  MULTIPLY  THE  SQUARE  OF  THE  DIAMETER  BY  -J-  OF  THE 

CIRCUMFERENCE. 

Or, 

3.  MULTIPLY  THE  SURFACE  BY  %  OF  THE  DIAMETER. 

1.  A  sphere  is  two-thirds  of  its  circumscribing  cylinder. 
(Sup.  Euc.  21.  3.)*     The  height  and  diameter  of  the  cylin- 
der are  each  equal  to  the  diameter  of  the  sphere.    The  solid- 
ity of  the  cylinder  is  equal  to  its  height  multiplied  into  the 
area  of  its  base,  (Art.  64.)  that  is  putting  D  for  the  diam- 
eter, 

DxD8X.?854     or     D8X.V854. 

And  the  solidity  of  the  sphere,  being  f  of  this,  is 
D8X.5236. 

2.  The  base  of  the  circumscribing  cylinder  is  equal  to  half 
the  circumference  multiplied  into  half  the  diameter ;  (Art. 
80.)  that  is,  if  C  be  put  for  the  circumference, 

-J-CxD  ;  and  the  solidity  is  -J-CxD1. 

Therefore,  the  solidity  of  the  sphere  is 

-f  ofiCxD'=D'XiC. 
8.  In  the  last  expression,  which  is  the  same  as  CxDx|D, 

*  Thomson's  Legendre,  12.  8. 


68  MENSURATION    OF   THE    SPHERE. 

we  may  substitute  S,  the  surface,  for  C  X  D.  (Art.  69.)    We 
then  have  the  solidity  of  the  sphere  equal  to 


Or,  the  sphere  may  be  supposed  to  be  filled  with  small 
pyramids,  standing  on  the  surface  of  the  sphere,  and  having 
their  common  vertex  in  the  centre.  The  number  of  these 
may  be  such,  that  the  difference  between  their  sum  and  the 
sphere  shall  be  less  than  any  given  quantity.  The  solidity 
of  each  pyramid  is  equal  to  the  product  of  its  base  into  £ 
of  its  height.  (Art.  48.)  The  solidity  of  the  whole,  there- 
fore, is  equal  to  the  product  of  the  surface  of  the  sphere 
into  \  of  its  radius,  or  •£•  of  its  diameter. 

71.  The  numbers  3.14159,  .7854,  .5236,  should  be  made 
perfectly  familiar.  The  first  expresses  the  ratio  of  the 
circumference  of  a  circle  to  the  diameter  ;  (Art.  23.)  the 
second,  the  ratio  of  the  area  of  a  circle  to  the  square  of  the 
diameter  (Art.  30.)  ;  and  the  third,  the  ratio  of  the  solidity 
of  a  sphere  to  the  cube  of  the  diameter.  The  second  is  -J- 
of  the  first,  and  the  third  is  £  of  the  first. 

As  these  numbers  are  frequently  occurring  in  mathemat- 
ical investigations,  it  is  common  to  represent  the  first  of  them 
by  the  Greek  letter  n.  According  to  this  notation, 

7r=3.14159,        }" 


If  D=the  diameter,  and  R=the  radius  of  any  circle  01 
sphere  ; 

Then,     D=2R    Da=4R2     D3=8R3. 


.  =the 

Or,  27tR  J  or  nR3  $      the  circ.  or  f^R'  $ 

solidity  of  the  sphere. 

Ex  1.  What  is  the  solidity  of  the  earth,  if  it  be  a  sphere 
7930  miles  in  diameter  ? 

Ans.  261,107,000,000  cubic  miles. 


MENSURATION    OF   THE    SPHERE.  69 

2.  How  many  wine  gallons  will  fill  a  hollow  sphere  4  feet 
in  diameter  ? 

Ans.  The  capacity  is  33.5104  feet=250f  gallons. 

3.  If  the  diameter  of  the  moon  be  2180  miles,  what  is  its 
solidity?  Ans.  5,424,600,000  miles. 

72.  If  the  solidity  of  a  sphere  be  given,  the  diameter  may 
be  found  by  reversing  the  first  rule  in  the  preceding  article ; 
that  is,  dividing  by  .5236  and  extracting  the  cube  root  of  the 
quotient. 

Ex.  1 .  What  is  the  diameter  of  a  sphere  whose  solidity  is 
65.45  cubic  feet  ?  Ans.  5  feet. 

2.  What  must  be  the  diameter  of  a  globe  to  contain  16755 
pounds  of  water?  Ans.  8  feet. 

PROBLEM  IX. 

To  find  the  CONVEX  SURFACE  of  a  SEGMENT  or   ZONE  of  a 
sphere. 

73.  MULTIPLY   THE    HEIGHT   OF   THE    SEGMENT    OR  ZONK 
INTO  THE  CIRCUMFERENCE  OF  THE  SPHERE. 

For  the  demonstration  of  this  rule,  see  Art.  69. 

Ex.  1.  If  the  earth  be  considered  a  perfect  sphere  7930 
miles  in  diameter,  and  if  the  polar  circle  be  23°  28'  from  the 
pole,  how  many  square  miles  are  there  in  one  of  the  frigid 
zones  ? 

If  PQOE  be  a  meridian  on  the 
earth,  ADB  one  of  the  polar  circles, 
and  P  the  pole ;  then  the  frigid  zone 
is  a  spherical  segment  described  by 
the  revolution  of  the  arc  APB  about 
PD.  The  angle  ACD  subtended  by 
the  arc  AP  is  23°  28'.  And  in  the 
right  angled  triangle  ACD, 


70  MENSURATION    OF   THE    SPHERE. 

R  :  AC  : :  cos  ACD  :  CD=3637. 

Then,  CP — CD =3965 — 3637=328=PD    the   height  of 
the  segment. 

And  328X7930X3.14159=8171400  the  surface. 

2.  If  the  diameter  of  the  earth  be  7930  miles,  what  is  the 
surface  of  the  torrid  zone,  extending 

23°  28'  on  each  side  of  the  equator  ? 

If  EQ  be  the  equator,  and  GH  one 
of  the  tropics,  then  the  angle  ECG  is 
23°  28'.  And  in  the  right  angled 
triangle  GCM, 

R  :  CG  :  :  sin  ECG  :  GM=CN=1578.9  the  height  of 
half  the  zone. 

The  surface  of  the  whole  zone  is  78669700. 

3.  What  is  the  surface  of  each  of  the  temperate  zones  ? 

The  height  DN=CP— CN— PD  = 2058.1* 
And  the  surface  of  the  zone  is  51273000. 

The  surface  of  the  two  temperate  zones  is  102,546,000 

of  the  two  frigid  zones  16,342,800 

of  the  torrid  zone  78,669,700 

of  the  whole  globe  197,558,500 

PROBLEM  X. 
To  find  the  SOLIDITY  of  a  spherical  SECTOR. 

74.    MULTIPLY    THE    SPHERICAL    SURFACE   BY    %   OF  THK 
EAJDIUS  OF  THE  SPHERE. 

The  spherical  sector  produced  by  the  revolution  of  ACBD 


MENSURATION    OF    THE    SPHEBB. 


71 


about  CD,  may  be  supposed  to  be  filled 

with  small  pyramids,   standing  on  the 

spherical  surface  ADB,  and  terminating 

in  the  point  C.     Their  number  may  be 

so  great,  that  the  height  of  each  shall 

differ    less   than    by  any  given    length 

from  the  radius  CD,  and  the  sum  of  their 

bases  shall  differ  less  than  by  any  given     .- 

quantity  from  the  surface  ABD.     The 

solidity  of  each  is  equal  to  the  product  of  its  base  into  %  of 

the  radius  CD.  (Art.  48.)     Therefore,  the  solidity  of  all  of 

them,  that  is,  of  the  sector  ADBC,  is  equal  to  the  product 

of  the  spherical  surface  into  $  of  the  radius. 

Ex.  Supposing  the  earth  to  be  a 
sphere  7930  miles  in  diameter,  and 
the  polar  circle  ADB  to  be  23°  28' 
from  the  pole ;  what  is  the  solidity  of 
the  spherical  sector  ACBP  ? 

Ans.  10,799,867,000  miles. 


PROBLEM  XL 
To  find  the  SOLIDITY  of  a  spherical  SEGMENT. 

75.  MULTIPLY  HALF  THE  HEIGHT  OF  THE  SEGMENT  INTO 
THE  AREA  OF  THE  BASE,  AND  THE  CUBE  OF  THE  HEIGHT 
INTO  .5236  ;  AND  ADD  THE  TWO  PRODUCTS. 

As  the  circular  sector  AOBC  consists  of  two  parts,  the 
segment  AOBP  and  the  triangle 
ABC ;  (Art.  35.)  so  the  sphericul 
sector  produced  by  the  revolution  of 
AOC  about  OC  consists  of  two  parts, 
the  segment  produced  by  the  revolu- 
tion of  AOP,  and  the  cone  produced 
by  the  revolution  of  AGP.  If  then 


MENSURATION    OF   THE    SPHERE. 


the  cone  be  subtracted  from  the  sec- 
tor, the  remainder  will  be  the  seg- 
ment. 

Let  CO=R,  the  radius  of  the  sphere, 
PB=r,  the  radius  of  the  base  of 

the  segment. 

P0=h,  the  height  of  the  segment, 
Then  PC=R — h,  the  axis  of  the  cone. 


The  sector =27rRxfcXiR( Arts.  71,  73,  74.)  = 
The  cone^^xi  (R— A) (Arts.  71,  66.)=in 

9- 

Subtracting  the  one  from  the  other, 

The  segment  =$nhR* — %ni*R.+\nki*. 

But  DOxPO=B03  (Trig.  97.*)=PO~a+PB2  (Euc.  47.  1.) 
That  is,  2R£=#'+r3.     So  that,  R=^!±!l 


Substituting  then,  for  R  and  R2,  then*  values,  and  multi- 
plying the  factors, 


The 


Which,  by  uniting  the  terms,  becomes 


The  first  term  here  is  i&X7^8,  half  the  height  of  the  seg- 
ment multiplied  into  the  area  of  the  base  ;  (Art.  71.)  and  the 
other  A'Xi7*,  the  cube  of  the  height  multiplied  into  .5236. 


*  Euclid  31,  3,  and  8,  6.   Cor. 


MENSURATION    OF    THE    SPHERE.  78 

If  the  segment  be  greater  than  a  hemisphere,  as  ABD ; 
the  cone  ABC  must  be  added  to  the  sector  ACBD. 

Let  PD=Athe  height  of  the  segment, 
Then  PC=h — R  the  axis  of  the  cone. 

The  sector  ACBD=f7r7*Ra 
The  cone=7rr2Xi(&— R)=i^r2 — ^ir'R 
Adding  them  together,  we  have  as  before, 
The  segment  =$nhR> — %nr*R+$nhr9. 

Cor.  The  solidity  of  a  spherical  segment  is  equal  to  half  a 
cylinder  of  the  same  base  and  height  -f-  a  sphere  whose 
diameter  is  the  height  of  the  segment.  For  a  cylinder  is 
equal  to  its  height  multiplied  into  the  area  of  its  base  ;  and 
a  sphere  is  equal  to  the  cube  of  its  diameter  multiplied  by 
.5236. 

Thus,  if  Oy  be  half  Ox,  the  spher- 
ical segment  produced  by  the  revo- 
lution, of  Oxt  is  equal  to  the  cylin- 
der produced  by  tvyx  -f-  the  sphere 
produced  by  Oyxz  ;  supposing  each 
to  revolve  on  the  line  Ox. 


Ex.  1.  If  the  height  of  a  spherical  segment  be  8  feet,  and 
the  diameter  of  its  base  25  feet ;  what  is  the  solidity  ? 

Ans.  (25)aX.'7854X4+88X.5236=2231.58  feet. 

2.  If  the  earth  be  a  sphere  7930  miles  in  diameter,  and  the 
polar  circle  23°  28'  from  the  pole,  what  is  the  solidity  of 
one  of  the  frigid  zones  ?  Ans.  1,303,000,000  miles. 

4 


MENSURATION    OF   THE    SPHERE. 


PROBLEM  XII. 
To  find  the  SOLIDITY  of  a  spherical  ZONE  or  frustum. 

76.  FROM  THE  SOLIDITY  OF  THE  WHOLE  SPHERE,  SUB- 
TRACT THE  TWO  SEGMENTS  ON  THE  SIDES  OF  THE  ZONE. 

Or, 

ADD  TOGETHER  THE  SQUARES  OF  THE  RADII  OF  THE  TWO 
ENDS,  AND  -J-  THE  SQUARE  OF  THEIR  DISTANCE  ;  AND  MULTIPLY 
THE  SUM  BY  THREE  TIMES  THIS  DISTANCC,  AND  -THE  PRODUCT 
BY  .5236. 

If  from  the  whole  sphere,  there 
be  taken  the  two  segments  ABP  and 
GHO,  there  will  remain  the  zone 
or  frustum  ABGH. 

Or,  the  zone  ABGH  is  equal  to 
the  difference  between  the  segments 
GHP  and  ABP. 


Let  NP=H 


TT    ,» 

~~      J  the  heights  of  the  two  segments, 
the  radii  of  their  bases. 


AD=r 

DN=d=H  —  h  the  distance  of  the  two  bases,  or  the 
height  of  the  zone. 

Then  the  larger  segment=£7iHR2+i-7iH3  )  /Arfc  *5  ^ 
And  the  smaller  segment=^7rAra-|-^-7r  A3      ( 

Therefore  the  zone  ABGH=i-n;  (3HR3+H3—  3£ra—  h*) 
By  the  properties  of  the  circle,  (Euc.  35,  3.) 
ONxH=R2.     Therefore,  (ON+H)xH=R2+Ha 
Or,  OP=?L+51 


MENSURATION    OF   THE    SPHERE.  75 

In  the  same  manner,  OP=  —  21— 

h 


Therefore,  3Hx  (r2+^)=3Ax(Ra-f  H*.) 
Or,  3Hra+3H/i2-—  3ARa—  3MF  =  0.     (Alg.  178.) 


To  reduce  the  expression  for  the  solidity  of  the  zone  to 
the  required  form,  without  altering  its  value,  let  these  terms 
be  added  to  it  :  and  it  will  become 


f7(3HRa+3Hr'—  3ARa— 
Which  is  equal  to 

in  X  3(H—  h)  X  (R3+ra-H  (H—  A)2) 

Or,  as  in  equals  .5236  (Art.  71.)  and  H  —  h  equals  d, 

The  zone=.5236X3rfx(Ra+r2+irfa.) 

Ex.  1.  If  the  diameter  of  one  end  of  a  spherical  zone  is 
24  feet,  the  diameter  of  the  other  end  20  feet,  and  the  dis- 
tance of  the  two  ends,  or  the  height  of  the  zone  4  feet  ; 
what  is  the  solidity?  Ans.  1566.6  feet. 

2.  If  the  earth  be  a  sphere  7930  miles  in  diameter,  and 
the  obliquity  of  the  ecliptic  23°  28'  ;  what  is  the  solidity  of 
one  of  the  temperate  zones  ? 

Ans.  55,390,500,000  miles. 

3.  What  is  the  solidity  of  the  torrid  zone  ? 

Ans.  147,720,000,000  miles. 

The  solidity  of  the  two  temperate  zones  is  110,781,000,000 
of  the  two  frigid  zones  2,606,000,000 

of  the  torrid  zone  147,720,000,000 

of  the  whole  globe  261,107,000,000 

4.  What  is  the  convex  surface  of  a  spherical  zone,  whose 
breadth  is  4  feet,  on  a  sphere  of  25  feet  diameter  ? 


Y6  MENSURATION    OF   SOLIDS. 

5.  What  is  the  solidity  of  a  spherical  segment,  whose 
height  is  18  feet,  and  the  diameter  of  its  base  40  feet  ?     . 


PROMISCUOUS   EXAMPLES   OF   SOLIDS. 

Ex.  1.  How  much  water  can  be  put  into  a  cubical  vessel 
three  feet  deep,  which  has  been  previously  filled  with  cannon 
balls  of  the  same  size,  2,  4,  6,  or  9  inches  in  diameter,  regu- 
larly arranged  in  tiers,  one  directly  above  another  ? 

Ans.  96£  wine  gallons. 

2.  If  a  cone  or  pyramid,  whose  height  is  three  feet,  be 
divided  into  three  equal  portions,  by  sections  parallel  to  the 
base  ;  what  will  be  the  heights  of  the  several  parts  ? 

Ans.  24.961,  6.488,  and  4.551  ihches. 

3.  What  is  the  solidity  of  the  greatest  square  prism  which 
can  be   cut  from  a  cylindrical  stick  of   timber,    2  feet   6 
inches  in  diameter  and  56  feet  long  ?* 

Ans.  175  cubic  feet. 

4.  How  many  such  globes  as  the  earth  are  equal  in  bulk 
to  the  sun;  if  the  former  is  7930  miles  in  diameter,  and  the 
latter  890,000  ?  Ans.  1,413,678. 

*  The  common  rule  for  measuring  round  limber  is  to  multiply  the 
square  of  the  quarter-girt  by  the  length.  The  quarter-girt  is  one-fourth 
of  the  circumference.  This  method  does  not  give  the  whole  solidity.  It 
makes  an  allowance  of  about  one-fifth,  for  waste  in  hewing,  bark,  &c. 
The  solidity  of  a  cylinder  is  equal  to  the  product  of  the  height  into  the 
area  of  the  base. 

If  C=the  circumference,  and  rr=3.14159,  then  (Art.  31.) 

C2     /    C  \  2  /    C   \  2 
The  area  of  the  base= -=  (_)  =  (^) 

If  then  the  circumference  were  divided  by  3.545,  instead  of  4,  and  the 
quotient  squared,  the  area  of  the  base  would  be  correctly  found.  See 
noteB. 


MENSURATION    OF    SOLIDS.  77 

6.  How  many  cubic  feet  of  wall  are  there  in  a  conical 
tower  66  feet  high,  if  the  diameter  of  the  base  be  20  feet 
from  outside  to  outside,  and  the  diameter  of  the  top  8  feet ; 
the  thickness  of  the  wall  being  4  feet  at  the  bottom,  and  de- 
creasing regularly,  so  as  to  be  only  two  feet  at  the  top  ? 

Ans.  7188. 

6.  If  a  metallic  globe  filled  with  wine,  which  cost  as  much 
at  5  dollars  a  gallon,  as  the  globe  itself  at  20  cents  for  every 
square  inch  of  its  surface ;  what  is  the  diameter  of  the  globe  ? 

Ans.  55.44  inches. 

7.  If  the  circumference   of  the   earth   be  25,000  miles, 
what  must  be  the  diameter  of  a  metallic  globe,  which,  when 
drawn  into  a  wire  /0-  of  an  inch  in  diameter,  would  reach 
round  the  earth  ?  Ans.  15  feet  and  1  inch. 

8.  If  a  conical  cistern  be  3  feet  deep,  7£,  feet  in  diameter 
at  the  bottom,  and  5  feet  at  the  top  ;  what  will  be  the  depth 
of  a  fluid  occupying  half  its  capacity  ? 

Ans.  14.535  inches. 

9.  If  a  globe  20  inches  in  diameter,  be  perforated  by  a 
cylinder  16  inches  in  diameter,  the  axis  of  the  latter  passing 
through  the  centre  of  the  former ;  what  part  of  the  solidity, 
and  the  surface  of  the  globe,  will  be  cut  aWay  by  the  cyl- 
inder? 

Ans.  3284  inches  of  the  solidity,  and  502,655  of  the  surface. 

10.  What  is  the  solidity  of  the  greatest  cube  which  can 
be  cut  from  a  sphere  three  feet  in  diameter  ? 

Ans.  5i  feet. 

11.  What  is  the  solidity  of  a  conic  frustum,  the  altitude 
of  which  is  36  feet,  the  greater  diameter  16,  and  the  lesser 
diameter  8  ? 

12.  What  is  the  solidity  of  a  spherical  segment  4  feet 
high,  cut  from  a  sphere  16  feet  in  diameter  ? 


78  ISOPERIMETRf. 


SECTION     V. 

ISOPERIMETRY. 

ART.  77.  It  is  often  necessary  to  compare  a  number  of 
different  figures  or  solids,  for  the  purpose  of  ascertaining 
which  has  the  greatest  area,  within  a  given  perimeter,  or  the 
greatest  capacity  under  a  given  surface.  We  may  have  oc- 
casion to  determine,  for  instance,  what  must  be  the  form  of 
a  fort,  to  contain  a  given  number  of  troops,  with  the  least 
extent  of  wall ;  or  what  the  shape  of  a  metallic  pipe  to  con- 
vey a  given  portion  of  water,  or  of  a  cistern  to  hold  a  given 
quantity  of  liquor,  with  the  least  expense  of  materials. 

78.  Figures  which  have  equal  perimeters  are  called  Iso- 
perimeters.  When  a  quantity  is  greater  than  any  other  of  the 
same  class,  it  is  called  a  maximum.  A  multitude  of  straight 
lines,  of  different  lengths,  may  be  drawn  within  a  circle. 
But  among  them  all,  the  diameter  is  a  maximum.  Of  all 
sines  of  angles,  which  can  be  drawn  in  a  circle,  the  sine  of 
90°  is  a  maximum. 

When  a  quantity  is  less  than  any  other  of  the  same  class, 
it  is  called  a  minimum.  Thus,  of  all  straight  lines  drawn 
from  a  given  point  to  a  given  straight  line,  that  which  is  per- 
pendicular to  the  given  line  is  a  minimum.  Of  all  straight 
lines  drawn  from  a  given  point  in  a  circle,  to  the  circumfer- 
ence, the  maximum  and  the  minimum  are  the  two  parts  of 
the  diameter  which  pass  through  that  point.  (Euc.  7,  3.) 

In  isoperimetry,  the  object  is  to  determine,  on  the  one 
hand,  in  what  cases  the  area  is  a  maximum,  within  a  given 
perimeter ;  or  the  capacity  a  maximum,  within  a  given  sur- 
face :  and  .on  the  other  hand,  in  what  cases  the  perimeter  is 


ISOPERIMETRY.  79 

a  minimum  for  a  given  area,  or  the  surface  a  minimum,  for  a 
given  capacity. 

PROPOSITION  I. 

79.  An  ISOSCELES  TRIANGLE  has  a  greater  area  than  any 
scalene  triangle,  of  equal  base  and  perimeter. 

If  ABC  be  an  isosceles  trian- 
gle whose  equal  sides  are  AC  and  D/ 
BC  ;  and  if  ABC'  be  a  scalene  tri- 
angle on  the  same  base  AB,  and 
having  AC'  +  BC'  =  AC+BC; 
then  the  area  of  ABC  is  greater 
than  that  of  ABC'. 

Let   perpendiculars   be    raised 
from  each  end  of  the  base,  extend 
AC  to  D,  make  C'D'  equal  to  AC',  join  BD,  and  draw  CH 
and  C'H'  parallel  to  AB. 

As  the  angle  CAB= ABC,  (Euc.  5, 1.)  and  ABD  is  a  right 
angle,  ABC+CBD=CAB+CDB=ABC+CDB.  Therefore 
CBD=CDB,  so  that  CD=CB  ;  and  by  construction,  C'D'= 
AC'.  The  perpendiculars  of  the  equal  right  angled  triangles 
CHD  and  CHB  are  equal ;  therefore,  BH=£BD.  In  the 
same  manner,  AH'=iAD'.  The  line  AD=AC+BC=AC/ 
+BC'=D'C'4-BC'.  But  D'C'+BC'>BD'.  (Euc.  20,  1.) 
Therefore,  AD>BD' ;  BD>AD',  (Euc.  47,  1.)  and  *  BD> 
\  AD'.  But  iBD,  or  BH,  is  the  height  of  the  isosceles  tri- 
angle ;  (Art.  1.)  and  £AD'  or  AH',  the  height  of  the  scalene 
triangle  ;  and  the  areas  of  two  triangles  which  have  the  same 
base  are  as  their  heights.  (Art.  8.)  Therefore  the  area  of 
ABC  is  greater  than  that  of  ABC'.  Among  all  triangles, 
then,  of  a  given  perimeter,  and  upon  a  given  base,  the  isos 
celes  triangle  is  a  maximum. 

Cor.  The  isosceles  triangle  has  a  less  perimeter  than  any 
scalene  triangle  of  the  same  base  and  area.     The  triangle 


80 


ISOPERIMETRY. 


ABC'  being  less  than  ABC,  it  is  evident  the  perimeter  of  the 
former  must  be  enlarged,  to  make  its  area  equal  to  the  area 
of  the  latter. 


PROPOSITION  II. 

80.  A  triangle  in   which  two  given   sides  mcike  a  RIGHT 
ANGLE,  has  a  greater  area  than  any  triangle  in  which  the  same 
sides  make  an  oblique  angle. 

If  BC,  BC'  and  BC"  be  equal, 
and  if  BC  be  perpendicular  to 
AB  ;  then  the  right  angled  trian- 
gle ABC,  has  a  greater  area  than 
the  acute  angled  triangle  ABC',  or 
the  oblique  angled  triangle  ABC". 

Let  P'C'  and  PC"  be  perpen- 
dicular to  AP.  Then,  as  the 
three  triangles  have  the  same  base  AB,  their  areas  are  as 
their  heights ;  that  is,  as  the  perpendiculars  BC,  P'C',  and 
PC".  But  BC  is  equal  to  BC',  and  therefore  greater  than 
P'C'.  (Euc.  47.  1.)  BC  is  also  equal  to  BC",  and  therefore 
greater  than  PC". 

PROPOSITION  III. 

81.  If  all  the  sides  EXCEPT  ONE  of  a  polygon   be  given, 
the  area  will  be  the  greatest,  when  the  given  sides  are  so  dis- 
posed that  the  figure  may  be  INSCRIBED  IN  A  SEMICIRCLE,  of 
which  the  undetermined  side  is  the  diameter. 


If  the  sides  AB,  BC,  CD,  DE, 
be  given,  and  if  their  position 
be  such  that  the  area,  included 
between  these  and  another  side 
whose  length  is  not  determined, 
is  a  maximum  ;  the  figure  may 


ISOPERIMETRY.  81 

be  inscribed  in  a  semicircle,  of  which  the  undetermined  side 
AE  is  the  diameter. 

Draw  the  lines  AD,  AC,  EB,  EC.  By  varying  the  angle 
at  D,  the  triangle  ADE  may  be  enlarged  or  diminished,  with- 
out affecting  the  area  of  the  other  parts  of  the  figure.  The 
whole  area,  therefore,  cannot  be  a  maximum,  unless  this  tri- 
angle be  a  maximum,  while  the  sides  AD  and  ED  are  given. 
But  if  the  triangle  ADE  be  a  maximum,  under  these  con- 
ditions, the  angle  ADE  is  a  right  angle ;  (Art.  80.)  and 
therefore  the  point  D  is  in  the  circumference  of  a  circle,  of 
which  AE  is  the  diameter.  (Euc.  31,3.)  In  the  same  man- 
ner it  may  be  proved,  that  the  angles  ACE  and  ABE  are 
right  angles,  and  therefore  that  the  points  C  and  B  are  in 
the  circumference  of  the  same  circle. 

The  term  polygon  is  used  in  this  section  to  include  trian- 
gles, and  four-sided  figures,  as  well  as  other  right-lined 
figures. 

82.  The  area  of  a  polygon,  inscribed  in  a  semicircle,  in 
the  manner  stated  above,  will  not  be  altered  by  varying  the 
order  of  the  given  sides. 

The  sides  AB,  BC,  CD,  DE,  are  the  chords  of  so 
many  arcs.  The  sum  of  these  arcs,  in  whatever  order 
they  are  arranged,  will  evidently  be  equal  to  the  semicircum- 
ference.  And  the  segments  between  the  given  sides  and  the 
arcs  will  be  the  same  in  whatever  part  of  the  circle  they  are 
situated.  But  the  area  of  the  polygon  is  equal  to  the  area 
of  the  semicircle,  diminished  by  the  sum  of  these  segments. 

83.  If  a  polygon,  of  which  all  the  sides  except  one  are 
given,  be  inscribed  in  a  semicircle  whose  diameter  is  the  un- 
determined side  ;  a  polygon  having  the  same  given  sides, 
cannot  be  inscribed  in  any  other  semicircle  which  is  either 
greater  or  less  than  this,  and  whose  diameter  is  the  undeter- 
mined side. 

The  given  sides  AB,  BC,  CD,  DE,  are  the  chords  of  arcs 
whose  sum  is  180  degrees.  But  in  a  larger  circle,  each 

4* 


82  ISOPERIMETRY. 

would  be  the  chord  of  a  less  number  of  degrees,  and  there- 
fore the  sum  of  the  arcs  would  be  less  than  180°  :  and  in  a 
smaller  circle,  each  would  be  the  chord  of  a  greater  number 
of  degrees,  and  the  sum  of  the  arcs  would  be  greater  than 
180°. 

PROPOSITION  IV. 

84.  A  polygon  INSCRIBED  IN  A  CIRCLE  has  a  greater  area, 
than  any  polygon  of  equal  perimeter,  and  the  same  number  of 
sides,  which  cannot  be  inscribed  in  a  circle. 

If  in  the  circle  ACHF,   (Fig.  30.)  there  be  inscribed  a 
c 


polygon  ABCDEFG ;  and  if  another  polygon  dbcdefg  (Fig. 
31.)  be  formed  of  sides  which  are  the  same  in  number  and 
length,  but  which  are  so  disposed,  that  the  figure  cannot  be 
inscribed  in  a  circle;  the  area  of  the  former  polygon  is 
greater  than  that  of  the  latter. 

Draw  the  diameter  AH,  and  the  chords  DH  and  EH. 
Upon  de  make  the  triangle  deh  equal  and  similar  to  DEH, 
and  join  ah.  The  line  ah  divides  the  figure  abcdhefg  into  two 
parts,  of  which  one  at  least  cannot,  by  supposition,  be  in- 
scribed in  a  semicircle  of  which  the  diameter  is  AH,  nor  in 
any  other  semicircle  of  which  the  diameter  is  the  undeter- 
mined side.  (Art.  83.)  It  is  therefore  less  than  the  corres- 
ponding part  of  the  figure  ABCDHEFG.  (Art.  81.)  And 
the  other  part  of  abcdhefg  is  not  greater  than  the  correspond- 


ISOPERIMETRY.  83 

ing  part  of  ABCDHEFG.  Therefore,  the  whole  figure 
ABODHEFG  is  greater  than  the  whole  figure  abcdhefg.  If 
from  these  there  be  taken  the  equal  triangles  DEH  and  deh, 
there  will  remain  the  polygon  ABCDEFG  greater  than  the 
polygon  abcdefg. 

85.  A  polygon  of  which  all  the  sides  are  given  in  num- 
ber and  length,  cannot  be  inscribed  in  circles  of  different 
diameters.  (Art.  83.)  And  the  area  of  the  polygon  will  not 
be  altered  by  changing  the  order  of  the  sides.  (Art.  82.) 


PROPOSITION  V. 

86.  When  a  polygon  has  a  greater  area  than  any  other,  of 
the  same  number  of  sides,  and  of  equal  perimeter,  the  sides  are 
EQUAL. 

The  polygon  ABCDF  (Fig.  29.) 
cannot  be  a  maximum,  among  all 
polygons  of  the  same  number  of 
sides,  and  of  equal  perimeters,  un-  P 
less  it  be  equilateral.  For  if  any 
two  of  the  sides,  as  CD  and  FD, 
are  unequal,  let  CH  and  FH  be 
equal,  and  their  sum  the  same  as 
the  sum  of  CD  and  FD.  The 

isosceles  triangle  CHF  is  greater  than  the  scalene  triangle 
CDF  (Art.  79.);  and  therefore  the  polygon  ABCHF  is 
greater  than  the  polygon  ABCDF ;  so  that  the  latter  is  not 
a  maximum. 

PROPOSITION  VI. 

87.  A  REGULAR  POLYGON  has  a  greater  area    than   any 
other  polygon  of  equal  perimeter,  and  of  the  same  number  of 
itides. 


84  ISOPERIMETRY 

For,  by  the  preceding  article,  the  polygon  which  is  a  max- 
imum among  others  of  equal  perimeters,  and  the  same  num- 
ber of  sides,  is  equilateral,  and  by  Art.  84,  it  may  be  m- 
scribed  in  a  circle.     But  if  a  poly- 
gon inscribed  in  a  circle  is  equilat- 
eral, as  ABDFGH,  it  is  also  equian- 
gular.   For  the  sides  of  the  polygon 
are  the  bases  of  so  many  isosceles 
triangles,  whose   common  vertex  is 
the  centre   C.     The  angles  at  these 
bases  are  all  equal ;  and  two  of  them, 

as  AHC  and  GHC,  are  equal  to  AHG  one  of  the  angles  of 
the  polygon.  The  polygon,  then,  being  equiangular,  as  well 
as  equilateral,  is  a  regular  polygon.  (Art.  1.  Def.  2.) 

Thus  an  equilateral  triangle  has  a  greater  area,  than  any 
other  triangle  of  equal  perimeter.  And  a  square  has  a 
greater  area  than  any  other  four-sided  figure  of  equal  pe- 
rimeter. 

Cor.  A  regular  polygon  has  a  less  perimeter  than  any 
other  polygon  of  equal  area,  and  the  same  number  of 
sides. 

For  if,  with  a  given  perimeter,  the  regular  polygon  is 
greater  than  one  which  is  not  regular ;  it  is  evident  the  pe- 
rimeter of  the  former  must  be  diminished,  to  make  its  area 
equal  to  that  of  the  latter. 


PROPOSITION  VII. 

88.  If  a  polygon  be  DESCRIBED  ABOUT  A  CIRCLE,  the  areas 
of  the  two  figures  are  as  their  perimeters. 

Let  ST  be  one  of  the  sides  of  a  polygon,  either  regular  or 


ISOPERIMETRT.  85 

not,  which  is  described  about  the  cir- 
cle LNR.  Join  OS  and  OT,  and 
to  the  point  of  contact  M  draw  the 
radius  OM,  which  will  be  perpen- 
dicular to  ST.  (Euc.  18,  3.)  The 
triangle  OST  is  equal  to  half  the 
base  ST  multiplied  into  the  radius 
OM.  (Art.  8.)  And  if  lines  be 
drawn,  in  the  same  manner,  from 
the  centre  of  the  circle,  to  the  extremities  of  the  sev- 
eral sides  of  the  circumscribed  polygon,  each  of  the  trian- 
gles thus  formed  will  be  equal  to  half  its  base  multiplied 
into  the  radius  of  the  circle.  Therefore  the  area  of  the 
whole  polygon  is  equal  to  half  its  perimeter  multiplied  into 
the  radius  :  and  the  area  of  the  circle  is  equal  to  half  its  cir- 
cumference multiplied  into  the  radius.  (Art  30.)  So  that 
the  two  areas  aie  to  each  other  as  their  perimeters. 

Cor.  1.  If  different  polygons  are  described  about  the 
same  circle,  their  areas  are  to  each  other  as  their  perimeters. 
For  the  area  of  each  is  equal  to  half  its  perimeter,  multi- 
plied into  the  radius  of  the  inscribed  circle. 

Cor.  2.  The  tangent  of  an  arc  is  always  greater  than  the 
arc  itself.  The  triangle  OMT  is  to  OMN,  as  MT  to  MN. 
But  OMT  is  greater  than  OMN,  because  the  former  includes 
the  latter.  Therefore,  the  tangent  MT  is  greater  than  the 
arc  MN. 

PROPOSITION  VIII. 

89.  A  CIRCLE  has  a  greater  area  than  any  polygon  of  equal 
perimeter. 

If  a  circle  and  a  regular  polygon  have  the  same  centre, 
and  equal  perimeters ;  each  of  the  sides  of  the  polygon 
must  fall  partly  within  the  circle.  For  the  area  of  a  drcum- 


86 


ISOPERIMETRY. 


scribing  polygon  is  greater  than  the  area  of  the  circle,  as 
the  one  includes  the  other  :  and  therefore,  by  the  preceding 
article,  the  perimeter  of  the  former  is  greater  than  that  of 
the  latter. 

Let  AD  then  be  one  side  of  a 
regular  polygon,  whose  perimeter 
is  equal  to  the  circumference  of 
the  circle  RLN.  As  this  falls 
partly  within  the  circle,  the  per- 
pendicular OP  is  less  than  the 
radius  OR.  But  the  area  of  the 
polygon  is  equal  to  half  its  pe- 
rimeter multiplied  into  this  per- 
pendicular (Art.  15.)  ;  and  the  area  of  the  circle  is  equal  to 
half  its  circumference  multiplied  into  the  radius.  (Art.  30.) 
The  circle  then  is  greater  than  the  given  regular  polygon ; 
and  therefore  greater  than  any  other  polygon  of  equal  pe- 
rimeter. (Art.  87.) 

Cor.  1.  A  circle  has  a  less  perimeter,  than  any  polygon  of 
equal  area. 

Cor.  2.  Among  regular  polygons  of  a  given  perimeter, 
that  which  has  the  greatest  number  of  sides,  has  also  the 
greatest  area.  For  the  greater  the  number  of  sides,  the 
more  nearly  does  the  perimeter  of  the  polygon  approach  to 
a  coincidence  with  the  circumference  of  a  circle. 


PROPOSITION  IX. 

90.  A  right  PRISM  whose  bases  are  REGULAR  POLYGONS,  has 
a  less  surface  than  any  other  right  prism  of  the  same  solidity, 
the  same  altitude,  and  the  same  number  of  sides. 

If  the  altitude  of  a  prism  is  given,  the  area  of  the  base  is 
as  the  solidity  (Art.  43.) ;  and  if  the  number  of  sides  is 


IS  OPS  RIME  TRY".  87 

also  given,  the  perimeter  is  a  minimum  when  the  base  is  a 
regular  polygon.  (Art.  87.  Cor.)  But  the  lateral  surface  is 
as  the  perimeter.  (Art.  47.)  Of  two  right  prisms,  then, 
which  have  the  same  altitude,  the  same  solidity,  and  the 
same  number  of  sides,  that  whose  bases  are  regular  polygons 
has  the  least  lateral  surface,  while  the  areas  of  the  ends  are 
equal. 

Cor.  A  right  prism  whose  bases  are  regular  polygons  has 
a  greater  solidity,  than  any  other  right  prism  of  the  same 
surface,  the  same  altitude,  and  the  same  number  of  sides. 


PROPOSITION  X. 

91.  A  right  CYLINDER  has  a  less  surface  than  any  right 
prism  of  the  same  altitude  and  solidity. 

For  if  the  prism  and  cylinder  have  the  same  altitude  and 
solidity,  the  areas  of  their  bases  are  equal.  (Artr.  64.)  But 
the  perimeter  of  the  cylinder  is  less,  than  that  of  the  prism 
(Art.  89.  Cor.  1.) ;  and  therefore  its  lateral  surface  is  less, 
while  the  areas  of  the  ends  are  equal. 

Cor.  A  right  cylinder  has  a  greater  solidity,  than  any  right 
prism  of  the  same  altitude  and  surface. 


PROPOSITION  XI. 

92.  A  CUBE  has  a  less  surface  than  any  other  right  paral- 
lelopiped  of  the  same  solidity. 

A  parallelepiped  is  a  prism,  any  one  of  whose  faces  may- 
be considered  a  base.  (Art.  41.  Def.  I  and  V.)  If  these  are 
not  all  squares,  let  one  which  is  not  a  square  be  taken  for  a 
base.  The  perimeter  of  this  may  be  diminished,  without 
altering  its  area  (Art.  87.  Cor.) ;  and  therefore  the  surface 


88  ISOPERIMETRY. 

of  the  solid  may  be  diminished,  without  altering  its  altitude 
or  solidity.  (Art.  43,  47.)  The  same  may  be  proved  of 
each  of  the  other  faces  which  are  not  squares.  The  surface 
is  therefore  a  minimum,  when  all  the  faces  are  squares,  that 
is,  when  the  solid  is  a  cube. 

Cor.  A  cube  has  a  greater  solidity  than  any  other  right 
parallelepiped  of  the  same  surface. 

PROPOSITION  XII. 

93.  A  CUBE  has  a  greater  solidity  than  any  other  right  par- 
allelopiped,  the  sum  of  whose  length,  breadth  and  depth,  is  equal 
to  ike  sum  of  the  corresponding  dimensions  of  the  cube. 

The  solidity  is  equal  to  the  product  of  the  length,  breadth, 
and  depth.  If  the  length  and  breadth  are  unequal,  the 
solidity  may  be  increased,  without  altering  the  sum  of  the 
three  dimensions.  For  the  product  of  two  factors  whose 
sum  is  giveji,  is  the  greatest  when  the  factors  are  equal. 
(Euc.  27.  6.)  In  the  same  manner,  if  the  breadth  and 
depth  are  unequal,  the  solidity  may  be  increased,  without 
altering  the  sum  of  the  three  dimensions.  Therefore,  the 
solid  cannot  be  a  maximum,  unless  its  length,  breadth,  and 
depth  are  equal. 

PROPOSITION  XIII. 

94.  If  O,     PRISM    BE     DESCRIBED     ABOUT     A    CYLINDER,     the 

capacities  of  the  two  solids  are  as  their  surfaces. 

The  capacities  of  the  solids  are  as  the  areas  of  their  bases, 
that  is,  as  the  perimeters  of  their  bases.  (Art.  88.)  But  the 
lateral  surfaces  are  also  as  the  perimeters  of  the  bases. 
Therefore  the  whole  surfaces  are  as  the  solidities. 

Cor.  The  capacities  of  different  prisms,  described  about 
the  same  right  cylinder,  are  to  each  other  as  their  surfaces. 


ISOPERIMETRY.  80 


PROPOSITION  XIV. 

95.  A  right  cylinder  WHOSE  HEIGHT  is  EQUAL  TO  THE 
DIAMETER  OF  ITS  BASE  has  a  greater  solidity  than  any  other 
right  cylinder  of  equal  surface. 

Let  C  be  a  right  cylinder  whose  height  is  equal  to  the  di- 
ameter of  its  base  ;  and  C'  another  right  cylinder  having  the 
same  surface,  but  a  different  altitude.  If  a  square  prism  P 
be  described  about  the  former,  it  will  be  a  cube.  But  a 
square  prism  P'  described  about  the  latter  will  not  be  a  cube. 

Then  the  surfaces  of  C  and  P  are  as  their  bases  (Art.  47. 
and  88.) ;  which  are  as  the  bases  of  C'  and  P',  (Sup.  Euc. 
7,  1.);  so  that, 

mrfC  :  surfP : :  base  C  :  base  P  : :  base  C'  :  base  P' : :  surfC' : 
surfP'. 

But  the  surface  of  C  is,  by  supposition,  equal  to  the  sur- 
face of  C'.  Therefore,  (Alg.  395.)  the  surface  of  P  is  equal 
to  the  surface  of  P'.  And  by  the  preceding  article, 

wild  P  :  solid  C  :  :  surfP  :surfC  : :  surfP'  :  surfQ'  : :  solid 
P'  :  solid  C'. 

But  the  solidity  of  P  is  greater  than  that  of  P'.  (Art.  92. 
Cor.)  Therefore  the  solidity  of  C  is  greater  than  that  of  C'. 

Schol.  A  right  cylinder  whose  height  is  equal  to  the  di- 
ameter of  its  base,  is  that  which  circumscribes  a  sphere.  It 
is  also  called  Archimedes'  cylinder ;  as  he  discovered  the 
ratio  of  a  sphere  to  its  circumscribing  cylinder;  and  these 
are  the  figures  which  were  put  upon  his  tomb. 

Cor.  Archimedes'  cylinder  has  a  less  surface,  than  any 
other  right  cylinder  of  the  same  capacity. 


90  ISOPflRIMETRY. 


PROPOSITION  XV. 

96.  If  a  SPHERE  BE  CIRCUMSCRIBED  by  a  solid  bounded  by 
plane  surfaces  ;  the  capacities  of  the  two  solids  are  as  their 
surfaces. 

If  planes  be  supposed  to  be  drawn  from  the  centre  of  the 
sphere,  to  each  of  the  edges  of  the  circumscribing  solid, 
they  will  divide  it  into  as  many  pyramids  as  the  solid  has 
faces.  The  base  of  each  pyramid  will  be  one  of  the  faces  ; 
and  the  height  will  be  the  radius  of  the  sphere.  The 
capacity  of  the  pyramid  will  be  equal,  therefore,  to  its  base 
multiplied  into  ^  of  the  radius  (Art.  48.);  and  the  capacity 
of  the  whole  circumscribing  solid,  must  be  equal  to  its  whole 
surface  multiplied  into  ^  of  the  radius.  But  the  capacity  of 
the  sphere  is  also  equal  to  its  surface  multiplied  into  •£  of  its 
radius.  (Art.  70.) 

Cor.  The  capacities  of  different  solids  circumscribing  the 
same  sphere,  are  as  their  surfaces. 

PROPOSITION  XVI. 

97.  A  SPHERE  has  a  greater  solidity  than  any  regular  poly- 
edron  of  equal  surface. 

If  a  sphere  and  a  regular  polyedron  have  the  same  centre, 
and  equal  surfaces ;  each  of  the  faces  of  the  polyedron  must 
fall  partly  within  the  sphere.  For  the  solidity  of  a  circum- 
scribing solid  is  greater  than  the  solidity  of  the  sphere,  as 
the  one  includes  the  other :  and  therefore,  by  the  preceding 
article,  the  surface  of  the  former  is  greater  than  that  of  the 
latter. 

But  if  the  faces  of  the  polyedron  fall  partly  within  the 
sphere,  their  perpendicular  distance  from  the  centre  must  be 
less  than  the  radius.  And  therefore,  if  the  surface  of  the 


ISOPERIMETRY.  "01 

polyedron  be  only  equal  to  that  of  the  sphere,  its  solidity 
must  be  less.  For  the  solidity  of  the  polyedron  is  equal  to 
its  surface  multiplied  into  ^  of  the  distance  from  the  centre. 
(Art.  59.)  And  the  solidity  of  the  sphere  is  equal  to  its 
surface  multiplied  into  -£•  of  the  radius. 

Cor.  A  sphere  has  a  less  surface  than  any  regular  poly- 
edron of  the  same  capacity. 


APPENDII 


GAUGING     OF     CASKS. 

ART.  119.  GAUGING  is  a  practical  art,  which  does  not  ad- 
mit of  being  treated  in  a  very  scientific  manner.  Casks  are 
not  commonly  constructed  in  exact  conformity  with  any  reg- 
ular mathematical  figure.  By  most  writers  on  the  subject, 
however,  they  are  considered  as  nearly  coinciding  with  one 
of  the  following  forms  : 

1.  )  C  of  a  spheroid, 

The  m,ddle  frustum  spind,e 


.     _  of  a  paraboloid, 

4      The  equal  frustums 


The  second  of  these  varieties  agrees  more  nearly  than  any 
of  the  others,  with  the  forms  of  casks,  as  they  are  com- 
monly made.  The  first  is  too  much  curved,  the  third  too 
little,  and  the  fourth  not  at  all,  from  the  head  to  the  bung. 

120.  Rules  have  already  been  given,  for  finding  the  capa- 
city of  each  of  the  four  varieties  of  casks.  (Arts.  68,  110, 
112,  118.)  As  the  dimensions  are  taken  in  inches,  these  rules 
will  give  the  contents  in  cubic  inches.  To  abridge  the  com- 
putation, and  adapt  it  to  the  particular  measures  used  in 
gauging,  the  factor  .7854  is  divided  by  282  or  231  ;  and 
the  quotient  is  used  instead  of  .7854,  for  finding  the  capa- 
city in  ale  gallons  or  wine  gallons. 


GAUGING.  98 

Now' =,002785,  or  .0028  nearly  ; 

282 

And --^=.0034 
231 

If  then  .0028  and  .0034  be  substituted  for  .7854,  in  the 
rules  referred  to  above ;  the  contents  of  the  cask  will  be 
given  in  ale  gallons  and  wine  gallons.  These  numbers  are 
to  each  other  nearly  as  9  to  11. 


PROBLEM  L 

To  calculate  the  contents  of  a  cask,  in  the  form  of  a  middle 
frustum  of  a  SPHEROID. 

121.  Add  together  the  square  of  the  head  diameter,  and 
twice  the  square  of  the  bung  diameter  :  multiply  the  sum  by 
•£  of  the  length,  and  the  product  by  ,0028  for  ale  gallons,  or 
by  .0034  for  wine  gallons, 

If  D  and  d=ihe  two  diameters,  and  Z=the  length; 
The  capacity  in  inches=(2D2+c?2)x^X.7854.  (Art.  110.) 

And  by  substituting  .0028  or  ,0034  for  ,7854,  we  have 
the  capacity  in  ale  gallons  or  wine  gallons. 

Ex.  What  is  the  capacity  of  a  cask  of  the  first  form, 
whose  length  is  30  inches,  its  head  diameter  18,  and  its  bung 
diameter  24  ? 

Ans.  41.3  ale  gallons,  or  50.2  wine  gallons. 

PROBLEM  II. 

To  calculate  the  contents  of  a  cask,  in  the  form  of  the  mid- 
dle frustum  of  a  PARABOLIC  SPINDLE. 

122.  Add  together  the  square  of  the  head  diameter,  and 
twice  the  square  of  the  bung  diameter,  and  from  the  sum 


94  GAUGING. 

subtract  -f  of  the  square  of  the  difference  of  the  diameters ; 
multiply  the  remainder  by  i  of  the  length,  and  the  product 
by  .0028  for  ale  gallons,  or  .0034  for  wine  gallons. 

The  capacity  in  inches  =(2D2+cP— |  (D— </)2)Xi^X 
.7854.  (Art.  118.) 

Ex.  What  is  the  capacity  of  a  cask  of  the  second  form, 
whose  length  is  30  inches,  its  head  diameter  18,  and  its 
bung  diameter  24  ? 

Ans.   40.9  ale  gallons,  or  49.7  wine  gallons. 

PROBLEM  III. 

To  calculate  the  contents  of  a  cask,  in  the  form  of  two  equal 
frustums  of  a  PARABOLOID. 

123.  Add  together  the  square  of  the  head  diameter,  and 
the  square  of  the  bung  diameter  ;  multiply  the  sum  by  half 
tfoe  length,  and   the  product  by  .0028   for  ale  gallons,  or 
.0034  for  wine  gallons. 

The  capacity  in  inches  =(D2-j-6?2)xi?X-7854.  (Art.  112 
Cor.) 

Ex.  What  is  the  capacity  of  a  cask  of  the  third  form, 
whose  dimensions  are,  as  before,  30,  18,  and  24  ? 

Ans.  37.8  ale  gallons,  or  45.9  wine  gallons. 

PROBLEM  IV. 

To  calculate  the  contents  of  a  cask,  in  the  form  of  two  equal 
frustums  of  a  CONE. 

124.  Add  together  the  square  of  the  head  diameter,  the 
square  of  the  bung  diameter,  and  the  product  of  the  two 
diameters  ;  multiply  the  sum  by  •§•  of  the  length,   and  the 
product  by.0028  for  ale  gallons,  or  .0034  for  wine  gallons. 
The  capacity  in  inches=(D9+G?2+Dc?)X^X.7854.  (Art.  68.) 


GAUGING.  05 

Ex.  What  is  the  capacity  of  a  cask  of  the  fourth  form, 
whose  length  is  30,  and  its  diameters  18  and  24? 

Ans.  37.3  ale  gallons,  or  45.3  wine  gallons. 

125.  The  preceding  rules,  though  correct  in  theory,  are 
not  very  well  adapted  to  practice,  as  they  suppose  the  form 
of  the  cask  to  be  known.     The  two  following  rules,  taken 
from  Hut  ton's  Mensuration,   may  be  used  for  casks  of  the 
usual  forms.     For  the  first,  three  dimensions  are  required, 
the  length,  the  head  diameter,  and  the  bung  diameter.     It 
is  evident  that  no  allowance  is  made  by  this,  for  different 
degrees  of  curvature  from  the  head  to  the  bung.     If  the 
cask  is  more  or  less  curved  than  usual,  the  following  rule  is 
to  be  preferred,  for  which  four  dimensions  are  required,  the 
head  and  bung  diameters,  and  a  third  diameter  taken  in  the 
middle  between   the  bung  and  the  head.     For  the  demon- 
stration of  these  rules,  see  Button's  Mensuration,  Part  V. 
Sec.  2.  Ch.  5  and  7. 

PROBLEM  V. 

To  calculate  the  contents  of  any  common  cask,  from  THREE 
dimensions. 

126.  Add  together 

25  times  the  square  of  the  head  diameter, 

39  times  the  square  of  the  bung  diameter,  and 

26  times  the  product  of  the  two  diameters ; 
Multiply  the  sum  by  the  length,  divide  the  product  by  90, 

and  multiply  the  quotient  by  .0028  for  ale  gallons,  or  .0034 
for  wine  gallons. 

The  capacity  in  inches=(39  Da+25cP  +  26Drf)X  j_x.'7854. 

90 

Ex.  What  is  the  capacity  of  a  cask  whose  length  is  30 
inches,  the  head  diameter  18,  and  the  bung  diameter  24? 
Ans.  39  ale  gallons,  or  47i  wine  gallons. 


96  GAUGING. 


PROBLEM  VI. 

To  calculate  the  contents  of  a  cask  from  FOUR  dimensions,  the 
length,  the  head  and  bung  diameters,  and  a  diameter  taken 
in  the  middle  between  the  head  and  the  bung. 

127.  Add  together  the  squares  of  the  head  diameter,  of 
the  bung  diameter,  and  of  double  the  middle  diameter  ; 
multiply  the  sum  by  •£•  of  the  length,  and  the  product  by 
.0028  for  ale  gallons,  or  .0034  for  wine  gallons. 

If  D=the  bung  diameter,  c?==the  head  diameter,  m=the 
middle  diameter,  and  /=the  length  ; 

The  capacity  in  inches=(Da 


Ex.  What  is  the  capacity  of  a  cask,  whose  length  is  30 
inches,  the  head  diameter  18,  the  bung  diameter  24,  and  the 
middle  diameter  22£  ? 

Ans.  41  ale  gallons,  or  49f  wine  gallons. 

128.  In  making  the  calculations  in  gauging,  according  to 
the  preceding  rules,  the  multiplications  and  divisions  are  fre- 
quently performed  by  means  of  a  Sliding  Rule,  on  which 
are  placed  a  number  of  logarithmic  lines,  similar  to  those  on 
Gunter's  Scale.  See  Trigonometry,  Sec.  VI.,  and  Note  C. 
p.  149. 

Another  instrument  commonly  used  in  gauging  is  the 
Diagonal  Rod.  By  this,  the  capacity  of  a  cask  is  very  ex- 
peditiously  found,  from  a  single  dimension,  the  distance  from 
the  bung  to  the  intersection  of  the  opposite  stave  with  the 
head  ;  but  this  process  is  not  considered  sufficiently  accurate 
for  casks  of  a  capacity  exceeding  40  gallons.  The  measure 
is  taken  by  extending  the  rod  through  the  cask,  from  the 
bung  to  the  most  distant  part  of  the  head.  The  number  of 
gallons  corresponding  to  the  length  of  the  line  thus  found,  is 
marked  on  the  rod.  The  logarithmic  lines  on  the  gauging 


GAUGING.  97 

rod  are  to  be  used  in  the  same   manner,  as  on  the  sliding 
rule. 

ULLAGE    OF    CASKS. 

129.  When  a  cask  is  partly  filled,  the  whole  capacity  is 
divided,  by  the  surface  of  the  liquor,  into  two  portions ;  the 
least  of  which,  whether  full  or  empty,  is  called  the  ullage. 
In  finding  the  ullage,  the  cask  is  supposed  to  be  in  one  of 
two  positions  ;  either  standing,  with  its  axis  perpendicular  to 
the  horizon ;  or  lying,  with  its  axis  parallel  to  the  horizon. 
The  rules  for  ullage  which  are  exact,  particularly  those  for 
lying  casks,  are  too  complicated  for  common  use.  The  fol- 
lowing are  considered  as  sufficiently  near  approximations. 
See  Button's  Mensuration. 


PROBLEM  VII. 
To  calculate  the  ullage  of  a  STANDING  cask. 

130.  Add  together  the  squares  of  the  diameter  at  the  sur- 
face of  the  liquor,  of  the  diameter  of  the  nearest  end,  and 
of  double  the  diameter  in  the  middle  between  the  other  two ; 
multiply  the  sum  by  £  of  the  distance  between  the  surface 
and  the  nearest  end,  and  the  product  by  .0028  for  ale  gal- 
lons, or  .0034  for  wine  gallons. 

If  D=the  diameter  of  the  surface  of  the  liquor, 
rf=the  diameter  of  the  nearest  end, 
m=the  middle  diameter,  and 

Z=the  distance  between  the  surface  and  the  nearest  end ; 
The  ullage  in  inches=(Da+c?'+2^r)X^X.V854. 

Ex.  If  the  diameter  at  the  surface  of  the  liquor,  in  a  stand- 
ing cask,  be  32  inches,  the  diameter  of  the  nearest  end  24, 
the  middle  diameter  29,  and  the  distance  between  the  sur- 

5 


98  GAUGING. 

face  of  the  liquor  and  the  nearest  end  1 2 ;  what  is  the  ul- 
lage? Ans.  27-f-  ale  gallons,  or  33f  wine  gallons. 

PROBLEM  VIII. 
To  calculate  the  ullage  of  a  LYING  cask. 

131.  Divide  the  distance  from  the  bung  to  the  surface  of 
the  liquor,  by  the  whole  bung  diameter,  find  the  quotient  in 
the  column  of  heights  or  versed  sines  in  a  table  of  circular 
segments,  take  out  the  corresponding  segment,  and  multiply 
it  by  the  whole  capacity  of  the  cask,  and  the  product  by  1-J- 
for  the  part  which  is  empty. 

If  the  cask  be  not  half  full,  divide  the  depth  of  the  liquor 
by  the  whole  bung  diameter,  take  out  the  segment,  multiply, 
&c.,  for  the  contents  of  the  part  which  is  full. 

Ex.  If  the  whole  capacity  of  a  lying  cask  be  41  ale  gal- 
lons, or  49f  wine  gallons,  the  bung  diameter  24  inches  and 
the  distance  from  the  bung  to  the  surface  of  the  liquor  6 
inches ;  what  is  the  ullage  ? 

Ans.  7f  ale  gallons,  or  9£  wine  gallons. 


NOTES. 


NOTE  A.  p.  39. 

THE  term  solidity  is  used  here  in  the  customary  sense,  to 
express  the  magnitude  of  any  geometrical  quantity  of  three 
dimensions,  length,  breadth,  and  thickness  ;  whether  it  be  a 
solid  body,  or  a  fluid,  or  even  a  portion  of  empty  space.  This 
use  of  the  word,  however,  is  not  altogether  free  from  objec- 
tion. The  same  term  is  applied  to  one  of  the  general  prop- 
erties of  matter  ;  and  also  to  that  peculiar  quality  by  which 
certain  substances  are  distinguished  from  fluids.  There 
seems  to  be  an  impropriety  in  speaking  of  the  solidity  of  a 
body  of  water  )  or  of  a  vessel  which  is  empty.  Some  writers 
have  therefore  substituted  the  word  volume  for  solidity.  But 
the  latter  term,  if  it  be  properly  defined,  may  be  retained 
without  danger  of  leading  to  mistake. 

NOTE  B.  p.  76. 

The  following  simple  rule  for  the  solidity  of  round  timber, 
or  of  any  cylinder,  is  nearly  exact  : 

Multiply  the  length  into  twice  the  square  of  ^  of  the  circum* 
ference. 

If  C=the  circumference  of  a  cylinder; 


The  area  of  the  base  But  2(     )=j- 

It  is  common  to  measure  hewn  timber,  by  multiplying  the 
length  into  the  square  of  the  quarter-girt.     This  gives  ex- 


100  NOTES. 

actly  the  solidity  of  a  parallelopiped,  if  the  ends  are  squares. 
But  if  the  ends  are  parallelograms,  the  area  of  each  is  less 
than  the  square  of  the  quarter-girt.  (Euc.  27.  6.) 

Timber  which  is  tapering  may  be  exactly  measured  by  the 
rule  for  the  frustum  of  a  pyramid  or  cone  (Art.  50,  68.) ; 
or,  if  the  ends  are  not  similar  figures,  by  the  rule  for  a  pris- 
moid.  (Art.  55.)  But  for  common  purposes,  it  will  be  suf- 
ficient to  multiply  the  length  by  the  area  of  a  section  in  the 
middle  between  the  two  ends. 


TABLE 


OP 

LOGARITHMS  OF  NUMBERS 

FROM 

1    TO    10,000. 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0.000000 

26 

.414973 

51 

1.707570 

76 

.880814 

2 

0.301030 

27 

.431364 

52 

1.716003 

77 

.886491 

3 

0.477121 

28 

.447158 

53 

1.724276 

78 

.892095 

4 

0.602060 

29 

.462398 

54 

.732394 

79 

.897627 

5 

0.698970 

30 

.477121 

55 

.740363 

80 

.903090 

6 

0.778151 

31 

.491362 

56 

.748188 

81 

.908485 

7 

0.845098 

32 

.505150 

57 

.755875 

82 

.913814 

8 

0.903090 

33 

.518514 

58 

.763428 

83 

.919078 

9 

.954243 

34 

.531479 

59 

.770852 

84 

.924279 

10 

.000000 

35 

.544068 

GO 

.778151 

85 

.929419 

11 

.041393 

36 

.556303 

61 

.785330 

86 

.934498 

12 

.079181 

37 

.568202 

62 

.792392 

87 

.939519 

J3 

.113943 

38 

.579784 

03 

.799341 

88 

.944483 

14 

.146128 

39 

.591065 

64 

.806180 

89 

.949390 

15 

.176091 

40 

.602060 

ft> 

.812913 

90 

.954243 

16 

.204120 

41 

.612784 

66 

.819544 

91 

.959041 

17 

.230449 

42 

.623249 

67 

.826075 

92 

.963788 

18 

.255273 

43 

.633468 

68 

.832509 

93 

.968483 

19 

.278754 

44 

.643453 

69 

.838849 

94 

.973128 

20 

.301030 

45 

.653213 

70 

.845098 

95 

.977724 

21 

.322219 

46 

.662758 

71 

.851258 

96 

.982271 

22 

.342423 

47 

.672098 

72 

.857333 

97 

.986772 

23 

.361728 

48 

.681241 

73 

.863323 

98 

.991226 

24 

.380211 

49 

.690196 

74 

.869232 

99 

.995635 

25 

.397940 

50 

.698970 

75 

1.875061 

100 

2.000000 

N.  B.  In  the  following  table,  in  the  last  nine  columns  of  each  page, 
where  the  first  or  leading  figures  change  from  9's  to  O's,  points  or  dots 
are  introduced  instead  of  the  O's  through  the  rest  of  the  line,  to  catch 
the  eye,  and  to  indicate  that  from  thence  the  annexed  first  two  figures 
of  the  Logarithm  in  the  second  column  stand  in  the  next  lower  line. 


A    TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


N.  |   0|1|2|3     4    5   |  6  |   7  |   8  |  9  |  D. 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

432 

101 

4321 

4751 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

428 

102 

8600 

9026 

9451 

9876 

.300 

.724 

1147 

1570 

1993 

2415 

424 

103 

012837 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

419 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

.361 

.775 

416 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

412 

106 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

408 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

404 

103 

033424 

3826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

400 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

396 

110 

041393 

1787 

2182 

2576 

2969 

3362 

3755 

4148 

4540 

4932 

393 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

389 

112 

9218 

9606 

9993 

.380 

.766 

1153 

1538 

1924 

2309 

2694 

386 

113 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

382 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

.320 

379 

115 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

376 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

372 

117 

8186 

8557 

8928 

9298 

966b 

..38 

.407 

.776 

1145 

1514 

369 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

366 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

120 

079181 

9543 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2426 

360 

121 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

123 

9905 

.258 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

351 

124 

093422 

3772 

4122 

4471 

'1820 

5169 

5518 

5866 

6215 

6562 

349 

125 

6910 

7257 

7604 

7951 

8-29H 

8644 

8990 

9335 

9681 

..26 

346 

128 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

343 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

340, 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.253 

338fc 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

335 

130 

113943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

333 

131 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

.245 

330 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

133 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

..12 

323 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

321 

136 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

315 

138 

9879 

.194 

.508 

.822 

1136 

1450 

1763 

2076 

2389 

2702 

314 

139 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140 

146128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

309 

141 

9219 

9527 

9835 

.142 

.449 

.756 

1063 

1370 

1676 

1982 

307 

142 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

144 

8362 

8664 

8965 

9266 

9567 

9868 

.168 

.469 

.769 

1068 

301 

145 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

299 

146 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

148 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

151 

8977 

9264 

9552 

9839 

.126 

.413 

.699 

.985 

1272 

1558 

287 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

153 

4691 

4975 

52.59 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

154 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

..51 

281 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

279 

156 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

157 

5899 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

158 

8657 

8932 

9206 

9481 

9755 

..29 

.303 

.577 

.850 

1124 

274 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

272 

N.  |   0     1   |  2    3   |   4    5   |   6  |   7  |   8  |  9   |  D. 

A  TABLE  OF  LOGARITHMS  FROM  1  TO  10,000. 


N.j 

0 

1. 

2 

3 

4 

s 

i; 

7 

8 

! 

D. 

!(>() 

204120 

4391 

4663 

4934 

5204 

.r>475 

5746 

6016 

62S6 

6556 

271 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

269 

162 

9515 

9783 

..51 

.319 

.586 

.853 

1121 

1388 

1654 

1921 

267 

1G3 

212188 

2454 

27-20 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

105 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

166 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

21% 

2456 

261 

167 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

168 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

.193 

251  i 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

254 

171 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

172 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

..50 

.300 

250 

174 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

249 

175 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

176 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.176 

245 

178 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

243 

179 

2853 

30% 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

255273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

181 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

182 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

184 

4818 

5054 

5290 

5525 

5761. 

5996 

6232 

6467 

6702 

6937 

235 

185 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

%46 

9279 

234 

186 

9513 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

233 

187 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

36% 

3927 

232 

188 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

189 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

82% 

8525 

229 

190 

278754 

8982 

9211 

9439 

9667 

9895 

.123 

.351 

.578 

.806 

223 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

194 

7802 

8026 

82-19 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

195 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

196 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

197 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

199 

8853 

9071 

9289 

9507 

9725 

9943 

.161 

.378 

.595 

.813 

218 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

203 

7486 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213 

204 

9630 

9843 

..56 

.268 

.481 

.693 

.906 

1118 

1330 

1542 

212 

305 

311754 

19(i6 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

208 

8063 

8272 

8481 

8689 

8898 

9100 

9314 

9522 

9730 

9938 

208 

209 

320146 

0354 

0562 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

322210 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

211 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

212 

633G 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

...8 

.211 

203 

214 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

202 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202 

216 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5650 

6059 

6-260 

201 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

218 

8456 

8656 

HHf>.-> 

9054 

9253 

9451 

9650 

9849 

..47 

.246 

199 

2J9 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

198 

N. 

0 

1 

2 

3   I 

4 

5 

6  I 

7 

8  I 

9 

D. 

A   TABLE    OF    LOGARITHMS    FROM    1    TO    10.000. 


N.  |   0   |  1  |  2  |  3    4   |   5  |  6  |   7    8  |  9  |  D. 

220 

342423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

221 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

591:2 

6157 

196 

222 

.  6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

..54 

194  1 

1  224 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1790 

1989 

193  - 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

227 

6026 

6217 

6408 

6599 

6790 

G981 

'172 

7363 

7554 

7744 

191 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

229 

9835 

..25 

.215 

.404 

.593 

.783 

.972 

1161 

1350 

1539 

189 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

188 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188  | 

232 

5488 

5675 

5862 

G049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

238 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

234 

9216 

9401 

9587 

9772 

9958 

.143 

.328 

.513 

.698 

.883 

185 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

237 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

239 

8398 

8580 

8761 

8943 

9124 

930G 

9487 

9668 

9849 

..30 

181 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

181 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

243 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

244 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

245 

9166 

9343 

9520 

9698 

9875 

..51 

.228 

.405 

.582 

.759 

177 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

21(59 

2345 

2521 

176 

247 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

248 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

G025 

175 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

9154 

9328 

9501 

173 

251 

9674 

9847 

..20 

.19-2 

.365 

.538 

.711 

.883 

1056 

1228 

173 

252 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

253 

3121 

3292 

3464 

3(535 

3807 

3978 

4149 

4320 

4492 

4663 

171 

254 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

255 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

1(59 

257 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 

169 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

259 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

414973 

5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

261 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

263 

9956 

.121 

.286 

.451 

.616 

.781 

.945 

1110 

1275 

1439 

165 

264 

421604 

1768 

]i)33 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

266 

4882 

5045 

5208 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

163 

267 

6511 

6674 

6336 

6999 

7161 

7324 

7486 

76-48 

7811 

7973 

162 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

269 

9752 

9914 

..76 

.236 

.398 

.559 

.720 

.881 

1042 

1203 

161 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

272 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004 

159 

273 

6163 

6322 

6481 

6640 

6798 

6957 

7116 

7275 

7433 

7592 

159 

274 

7751 

7909 

8007 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

275 

9333 

9491 

9648 

9806 

9964 

.122 

.279 

.437 

.594 

.752 

158 

276 

440909 

1066 

HM4 

1381 

1.538 

1095 

1852 

2009 

2166 

2323 

157 

277 

2480 

2637 

2793 

29M 

3106 

3263 

3419 

3576 

3732 

3889 

157 

278 

4045 

4-201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

279 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

JNT|   0   |  1   |  2 

3|4|5|6|7|8|9|D.  | 

A  TABLE  OP  LOGARITHMS  PROM  1  TO  10,000. 


N.  |   0|l|2|3|4|5|6|-7    8  |  9  |  D. 

280 

447158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

155 

281 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

..95 

154 

282 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

284 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4(592 

153 

285 

4845 

4997 

5150 

530-2 

5454 

5606 

5758 

5910 

60(52 

6214 

152 

286 

6366 

6518 

6670 

6821 

G973 

7125 

7276 

7428 

7.r>7'.l 

7731 

152 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

288 

9392 

9543 

9694 

9845 

9995 

.146 

.296 

.447 

.597 

.748 

151 

289 

460898 

1048 

1198 

1348 

1499 

1649 

Ii99 

1948 

2098 

2248 

150 

290 

462398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

150 

201 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

903 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

295 

9822 

9969 

.116 

.263 

.410 

.557 

.704 

.851 

.998 

1145 

147 

296 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

297 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

298 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

299 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

145 

301 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

302 

480007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

144 

303 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

304 

2874 

3016 

3159 

33(1-2 

3445 

3587 

3730 

3872 

4015 

4157 

143 

305 

4300 

4442 

4585 

47-27 

4869 

5011 

5153 

5295 

5437 

5579 

142 

306 

5721 

5863 

•5005 

6147 

6289 

6430 

6572 

6714 

6855 

.6997 

142 

307 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

308 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

309 

9958 

..99 

.239 

.380 

.520 

.661 

.801 

.941 

1081 

1222 

140 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

311 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

312 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

313 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

314 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

315 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

316 

9687 

9824 

9962 

..99 

.236 

.374 

.511 

.648 

.785 

.922 

137 

317 

501059 

1196 

1333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

318 

2427 

2564 

2700 

2837 

2973 

3109 

3-246 

3382 

3518 

3655 

136 

319 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

505150 

5286 

5421 

5557 

5693 

5828 

5964 

6099 

6234 

6370 

136 

321 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

322 

7856 

7991 

81& 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

323 

9203 

9337 

9471 

9606 

9740 

9874 

...9 

.143 

.277 

.411 

134 

324 

510545 

0679 

0813 

0947 

1081 

1215 

1349 

1482 

1616 

1750 

134 

325 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

i  326 

3218 

3351 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4414 

133 

327 

4548 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

133 

32H 

5874 

6001) 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

132 

329 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

5ia->i4 

8646 

8777 

8909 

9040 

9171 

9303 

9434 

9566 

9697 

131 

331 

9828 

9959 

..90 

.221 

.353 

.484 

.615 

.745 

.876 

1007 

131 

332 

521138 

1269 

1400 

1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

335 

5045 

5174 

5304 

5434 

5563 

5693 

58S3 

5951 

6081 

6210 

129 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

737-2 

7501 

129 

337 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

129 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

..72 

128 

339 

530200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

1351 

128 

N.  |   0   |  1   |  2  |  3   |  4  |  5   |6|7    8  |  9   |  D. 

A   TABLE    OF    LOGARITHMS    FROM    1    TO    ]  0,000. 


N.    0     1 

2|3    4     5    6|7     8|9    I). 

340 

531479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

2501) 

-J&JV 

128 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

342 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

343 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

344 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

..79 

.204 

125 

347 

540329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1454 

125 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

340 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

544068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

124 

351 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

354 

9003 

9126 

9249 

9371 

9494 

9f>16 

9739 

9861 

9984 

.106 

123 

355 

550228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

357 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

359 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

556333 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

120 

361 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9(367 

9787 

120 

363 

9907 

..26 

.146 

.265 

.385 

.504 

.624 

.743 

.863 

.982 

119 

364 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

356 

3481 

3630 

3718 

3837 

3955 

4074 

4102 

4311 

4429 

4548 

119 

367 

4686 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118 

368 

5848 

5966 

6J84 

6232 

6320 

6437 

6555 

6673 

6791 

0909 

118 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

568202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

117 

371 

9374 

9491 

9608 

9725 

9842 

9959 

-.76 

.193 

•309 

.426 

117 

372 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

1476 

1592 

117 

373 

1709 

1825 

1942 

2358 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

38.K) 

3915 

116 

375 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

622(3 

115 

377 

6341 

6457 

6572 

6687 

6832 

6917 

7032 

7147 

7262 

7377 

115 

378 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

115 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

579784 

9898 

..12 

.126 

.241 

.355 

.469 

.583 

.697 

.811 

114 

381 

580925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

113 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5039 

5122 

5235 

5348 

113 

385 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112 

388 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9(315 

9726 

9838 

112 

389 

9950 

..61 

.173 

.284 

.396 

.507 

.619 

.730 

.842 

.953 

112 

390 

591065 

1178 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

111 

391 

2177 

2283 

2399 

2510 

2621 

2732 

2843 

2934 

3064 

3175 

111 

392 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

111 

393 

4393 

4503 

4614 

4724 

4834 

'4945 

5055 

5165 

5276 

5386 

110 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

397 

8791 

8900 

9309 

9119 

9228 

9337 

9446 

9556 

9565 

9774 

109 

398 

9883 

9992 

.10] 

.210 

.319 

.428 

.537 

.646 

.755 

.854 

109 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

109 

N.    0     1  |  2 

3   |   4  |   5     6(7 

8  |  9  |  D. 

A  TABLE  or  LOGARITHMS  FROM  1  TO  10,000. 


N.  |0|1|2|3|4|5    6  |   7     8  |  9    D. 

400 

6021X50 

2169 

2277 

2386 

2494 

2(503 

•2711 

2819 

2tS8 

303(5 

108 

401 

3144 

:{-2:>:i 

33(51 

34159 

3577 

3(5815 

3794 

3902 

4010 

4118 

108 

402 

42-26 

4334 

4442 

4.~).">0 

465H 

4760 

4874 

4982 

5089 

5197 

108 

403 

5305 

5413 

5521 

5'WH 

5736 

5844 

5951 

6059 

6166 

6274 

108 

404 

6381 

6489 

6f>9(5 

6704 

6811 

(UU'.I 

702I5 

7133 

7241 

7348 

107 

405 

7455 

75152 

76(59 

7777 

7884 

T'.I'.M 

8098 

8205 

8312 

8419 

107 

406 

8526 

81533 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

107 

407 

B594 

'J7.ll 

9808 

1)914 

..21 

.128 

.234 

.341 

.447 

.554 

107' 

408 

6106(50 

07(57 

0873 

0979 

108(5 

1192 

1298 

1405 

1511 

1617 

106 

41)9 

17-2:5 

1829 

19315 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

612784 

2890 

2998 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

106 

411 

3842 

3047 

4053 

4159 

42(54 

4370 

4475 

4581 

468(5 

4-;<)2 

100 

412 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5845 

105 

413 

5950 

6055 

6160 

6265 

6370 

647(5 

6581 

6686 

6790 

6895 

105 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

105 

415 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

416 

9093 

9198 

!)3()-2 

9406 

9511 

9615 

9719 

9824 

9928 

..32 

104 

417 

620136 

0240 

0344 

0448 

0552 

0658 

0760 

0864 

0968 

1072 

104 

418 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

104 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

104 

420 

623249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

103 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

422 

5312 

5415 

5518 

5(521 

5724 

5827 

5929 

6032 

6135 

6238 

103 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

103 

4-24 

7:«;ii 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

102 

425 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

426 

9410 

9512 

9613 

9715 

9817 

9919 

..21 

.123 

.224 

.326 

102 

427 

630428 

0530 

0631 

0733 

0835 

0936 

1038 

1139 

1241 

J342 

102 

428 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

101 

429 

2457 

2559 

2660 

2761 

2882 

2953 

3064 

3165 

3266 

3367 

101 

430 

633468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

100 

431 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

100 

4.i2 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

433 

6488 

6588 

61T88 

6789 

6889 

(5989 

7089 

7189 

7290 

7390 

100 

434 

7493 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8291) 

8389 

99 

435 

848J 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

99 

436 

948(5 

9586 

9686 

9785 

9885 

9984 

..84 

.183 

.283 

.382 

99 

437 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

438 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

99 

440 

r,i:u.-,3 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

98 

441 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

522(5 

.1324 

98 

442 

54-22 

5521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

6306 

98 

443 

6404 

6502 

0600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

444 

7383 

7481 

7579 

7676 

7774 

7H72 

7969 

8067 

8165 

8262 

98 

445 

8360 

8458 

8555 

8H53 

8750 

8848 

8945 

9043 

9140 

9237 

97 

4  tii 

9335 

9432 

9530 

9827 

9724 

9821 

9919 

..16 

.113 

.210 

97 

447 

650308 

0405 

0502 

0599 

0896 

0793 

0890 

0987 

1084 

1181 

97 

448 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

450 

653213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

96 

451 

4177 

4273 

43(59 

4465 

4562 

4658 

4754 

4850 

4946 

5042 

96 

452 

5138 

5235 

5331 

5427 

5523 

5819 

5715 

5810 

5906 

6002 

96 

453 

BOOS 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

96 

454 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

455 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

95 

456 

8955 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

95 

457 

9916 

..11 

.105 

.201 

.295 

.331 

.486 

.581 

.676 

.771 

95 

458 

660865 

0960 

1055 

1150 

1-245 

1339 

1434 

1529 

1623 

1718 

95 

459 

1813 

1907 

2002 

2098 

2191 

2286 

23HO 

2475 

2569 

2663 

95 

N.  |   0   |   1  |  2  |  3   j   4  I   5    6  |   7    8  |  9  |  D. 

A   TABLE    OF    LOGARITHMS   FROM    1    TO    10.000. 


N.  |   0     1|2J3|4|5|6|7|8|9|D. 

460 

662758 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3512 

3607 

94 

461 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

462 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

463 

5581 

5675 

5769 

5862 

5955 

6050 

6143 

6237 

6331 

64-24 

94 

.464 

6518 

6612 

0705 

6799 

6892 

0986 

7079 

7173 

7266 

7360 

94 

465 

7453 

7546 

7640 

7733 

782a 

7920 

8013 

8106 

8199 

8293 

93 

466 

838ti 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

407 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

..60 

.153 

93 

468 

670246 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

93 

469 

1173 

1265 

1358 

1451 

1513 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

2190 

2283 

2375 

2467 

2560 

2652 

2744 

2836 

2929 

92  , 

471 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

92 

472 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

92 

473 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

92 

474 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

475 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

751b 

91 

476 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

91 

477 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9155 

9246 

9337 

91 

478 

9428 

9519 

9610 

9700 

9791 

9882 

9D73 

..63 

.154 

.245 

91 

479 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

91 

480 

681241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

90 

481 

2145 

2235 

2326 

2416 

2506 

2598 

2686 

2777 

2867 

2957 

90 

482 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3917 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

485 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

486 

6036 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

488 

8420 

8509 

8598 

8ii87 

8776 

88W 

8953 

9042 

9131 

9220 

89 

489 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

..19 

.107 

89 

490 

690196 

0285 

0373 

0462 

0550 

0639 

0728 

0816 

0905 

0993 

89 

491 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

88 

492 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

88 

493 

2847 

2935 

3J23 

3111 

3199 

3287 

3375 

3463 

3551 

3(539 

88 

494 

3727 

3815 

3903 

399  1 

4078 

4166 

4254 

4342 

4430 

4517 

88 

495 

4505 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

496 

5482 

55;>9 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

498 

7229 

7317 

7401 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 

8101 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

87 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

87 

5.11 

9838 

9924 

..11 

..98 

.184 

.271 

.358 

.444 

.531 

.617 

87 

502 

700704 

0790 

0877 

01)63 

1050 

1136 

1222 

1309 

1395 

1482 

86 

503 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

504 

2431 

2517 

2603 

2>i89 

2775 

2861 

2947 

3033 

3119 

3205 

86 

505 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3895 

3979 

4065 

86 

506 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

86 

507 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

86 

508 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

6718 

680J3 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

85 

510 

707570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

85 

511 

8421 

8506 

8591 

8676 

8761 

884fi 

8931 

9015 

9100 

9185 

85 

512 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

..33 

85 

513 

710117 

0202 

0287 

0371 

0456 

0540 

0825 

0710 

0794 

0879 

85 

514 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

84 

515 

1807 

1892 

1976 

2060 

2144 

2-220 

2313 

2397 

2481 

2566 

84 

516 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3333 

3407 

84 

517 

3491 

3575 

3650 

374-2 

3826 

3910 

3994 

4078 

4162 

4246 

84 

518 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

84 

519 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

84 

IN.  |   0|1|2|3|4|5|6    7    8  |  9  |  D. 

A   TABLB    OF    LOGARITHMS    FROM    1    TO    10,000. 


N.  |   0     1     2  |  3     4     5  |  6     7  |   8    9    D. 

520 

716003 

6087 

6170 

6254 

6337 

6421 

r.:.o  t 

6588 

6671 

6754 

83 

521 

6838 

69-21 

701)4 

7088 

7171 

7254 

73,'W 

7421 

7504 

7587 

83 

522 

Ttn 

7754 

7837 

7920 

81)03 

8086 

8169 

P2.-.3 

8336 

8419 

83 

523 

6903 

8585 

8668 

8751 

8834 

8917 

90:K) 

SHKJ 

911)5 

9248 

83 

524 

9:01 

9414 

911)7 

9580 

9663 

9745 

9828 

9911 

9991 

..77 

83 

525 

72.)  159 

0242 

0325 

0407 

0490 

0573 

0655 

0738 

OH21 

0903 

83 

526 

0936 

1068 

1151 

1233 

1316 

1398 

1481 

J.-)!i3 

1(546 

1728 

82 

527 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

23ri7 

2469 

2552 

82 

528 

3634 

2716 

2798 

2881 

2;i;i:5 

3045 

3127 

3209 

'3291 

3374 

82 

529 

3456 

3538 

34BQ 

3702 

3784 

38156 

3943 

4030 

4112 

4194 

82 

530 

724276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5013 

82 

531 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

50(57 

5748 

5830 

82 

538 

5912 

5993 

6075 

6156 

(i23S 

63-20 

(i401 

6483 

6564 

6646 

82 

533 

0727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

534 

7.')  1  1 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

535 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

S922 

9003 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

537 

9974 

..55 

.136 

.217 

.293 

.378 

.459 

.540 

.621 

.702 

81 

538 

730782 

0863 

Oi)41 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

539 

1539 

1669 

1750 

1830 

1911 

1991 

«072 

2152 

2233 

2313 

81 

540 

732394 

2474 

2535 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

80 

541 

3197 

3278 

3353 

3438 

3518 

3598 

3679 

3759 

3839 

3919 

80 

542 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

80 

543 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5519 

80 

544 

5599 

5679 

5759 

5838 

5-)  18 

5998 

6078 

6157 

6237 

6317 

80 

545 

0397 

6476 

655!! 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

54(5 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

547 

7987 

8067 

8146 

8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

.  548 

8781 

88(50 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

549 

9572 

9651 

9731 

9810 

9839 

9968 

..47 

.126 

.205 

.284 

79 

550 

740363 

0412 

0521. 

0600 

0678 

0757 

0836 

0915 

0994 

1073 

79 

551 

1152 

1330 

1309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

79 

552 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2646 

79 

553 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

554 

3510 

3588 

3667 

3745 

3323 

3902 

3980 

4053 

4136 

4215 

78 

555 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

553 

5075 

5153 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

557 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

8634 

6712 

6790 

6868 

15945 

7023 

7101 

7179 

725(5 

7334 

78 

559 

7412 

7489 

7567 

7(545 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560 

748188 

8266 

8343 

8421 

8493 

8576 

8(153 

8731 

8808 

8885 

77 

5(51 

89153 

9940 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

562 

9738 

9814 

9891 

9968 

..45 

.123 

.200 

.277 

.354 

.431 

77 

563 

750508 

058(5 

0563 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

564 

1279 

1356 

1433 

1510 

1587 

1G64 

1741 

1818 

1895 

1972 

77 

565 

2048 

2125 

2203 

2279 

2356 

2433 

2509 

2586 

2663 

2740 

77 

566 

:2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430 

3506 

77 

5!>7 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

568 

4348 

4425 

4501 

4578 

4654 

4730 

4807 

4883 

4960 

5036 

76 

569 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

76 

570 

755875 

5951 

6027 

6103 

6180 

6256 

6332 

6408 

6484 

6560 

76 

571 

PQ38 

6712 

6788 

6864 

6940 

7016 

7092 

71(58 

7244 

7320 

76 

572 

7396 

7472 

7548 

7624 

7700 

777.-. 

7351 

7927 

8003 

8079 

76 

573 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761 

8836 

76 

574 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

575 

9668 

9743 

9819 

9894 

9970 

..45 

.121 

.198 

.272 

.347 

75 

576 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950 

1025 

1101 

75 

577 

1176 

1251 

1326 

1402 

1477 

1552 

1627 

1702 

1778 

1853 

75 

578 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2453 

2529 

2604 

75 

579 

2679 

27.->l 

3898 

2904 

2978 

31).-.:$ 

3128 

3203 

3278 

3353 

75 

N.  1   0   |   1    2  |  3   j   4  |   5  |   6    7  |   8  |  9  |  D. 

10 


A   TABLE    OF    LOGARITHMS    FROM    1    TO    10,000. 


N.  |   0   |  1  |  2    3|4     5    6  |   7  |   8  |  9  |  D.  j 

580 

763428 

3503 

3578 

3653 

3727 

3802  3-577 

31)52 

4027  1  4101 

75 

581 

4176 

4251 

4326 

4400 

4475 

4550 

4624 

4699 

4774 

4848 

75 

58-2 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

5445 

5520 

5594 

75 

583 

5669 

5743 

5818 

5892 

591)6 

6041 

6115 

6190 

6264 

6338 

74 

584 

6-113 

6487 

6562 

6636 

67JO 

6785 

6859 

6933 

7007 

7032 

74 

585 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675 

7749 

7823 

74 

586 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

8416 

8490 

8564 

74 

587 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

588 

9377 

9451 

9.525 

9599 

9673 

9746 

9820 

9894 

99G8 

..42 

74 

589 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

74 

590 

770852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

74 

591 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

73 

592 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

593 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

594 

3786 

3860 

3!)33 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

595 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

596 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

73 

597 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

73 

598 

6701 

6774 

C846 

69  U) 

6992 

TOM 

7137 

7209 

7282 

7354 

73 

599 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

600 

778151 

8224 

8296 

8368 

8441 

8.">  ]  3 

8585 

8658 

8730 

8802 

72 

601 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

602 

9596 

9669 

9741 

9813 

9885 

9957 

..29 

.101 

.173 

.245 

72 

603 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

604 

1037 

1109 

1J81 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

72 

605 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

606 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

6')7 

3189 

3260 

3332 

3-103 

3475 

35-16 

3618 

3689 

3761 

3832 

71 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

609 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

71 

611 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

612 

6751 

6822 

6893 

6964  7035 

7106 

7177 

7248 

7319 

731K) 

71 

613 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

80-27 

8098 

71 

614 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

615 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

616 

9581 

9851 

9722 

9792 

9863 

9933 

...4 

..74 

.144 

.215 

70 

617 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

70 

618 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

14cO 

1550 

1620 

70 

619 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

70 

622 

3790 

3860 

3930 

400!) 

4070 

4139 

4209 

4279 

4349 

4418 

70 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

70 

624 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

70 

625 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

626 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

69 

627 

7268 

7337 

7406 

7475 

7545 

7614 

7683 

7752 

7821 

7890 

69 

628 

7960 

8029 

8098 

8167 

8236 

8305 

8374 

8443 

8513 

8582 

69 

629 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

799341 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

69 

631 

800029 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

0580 

0648 

69 

632 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

69 

633 

1404 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

69 

634 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2837 

2705 

69 

635 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

68 

636 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

3935 

4003 

4071 

68 

637 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

(58 

638 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

639    5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

68 

N.  |   0|l|2|3j4|5 

6  |  7  |  8  |  9   D. 

A    TAET,E    OF    LOGARITHMS    FROM    1    TO    10,000. 


11 


N.    0   |   1   |  2  |  3     4   |   5  L  6  |   7  |   8  |  9  |  D. 

640* 

rHIOIsil 

8948 

6316 

•  5384 

o-ir.i 

0519 

df>87 

i;t;.-,r> 

07-23 

6790 

68 

641 

6858 

osi-20 

0994 

7001 

7128 

7197 

7264 

7332 

7400 

7467 

68 

64-2 

::>:(.-> 

7003 

7670 

7738 

780fl 

7873 

7941 

8008 

8076 

8143 

68 

643 

8211 

827!) 

8346 

8414 

H48  1 

8549 

8016 

8084 

8751 

8818 

67 

644 

8886 

8953 

9021 

9068 

9156 

9-2-23 

9290 

9358 

9425 

9492 

67 

645 

9.">00 

9027 

9094 

970-2 

9829 

9890 

9904 

..31 

-.98 

.165 

67  1 

64S 

810233 

0300 

0367 

0434 

0501 

0509 

0030 

0703 

0770 

0837 

67  1 

647 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

648 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

649 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

67 

650 

812913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

67 

651 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

67 

652 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

653 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

66 

654 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

655 

6241 

6308 

6374 

6440 

6506 

6573 

6039 

6705 

6771 

6838 

66 

656 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

657 

7565 

7631 

7698 

7764 

7830 

7896 

7902 

8028 

8094 

8160 

66 

658 

8220 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819544 

9610 

9676 

9741 

9807 

9873 

9939 

...4 

..70 

.136 

66 

P$l 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0061 

0727 

0792 

66 

b62 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

663 

1514 

1579 

1045 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

1  664 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2620 

2691 

2756 

65 

665 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

G06 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3990 

4001 

65 

667 

4126 

4191 

425(5 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

668 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

529(5 

5361 

65 

609 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

65 

670 

826075 

OHO 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

65 

671 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

65 

07-2 

7359 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

673 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

675 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

64 

676 

9947 

..11 

..75 

.139 

.204 

.208 

.332 

.390 

.460 

.525 

64 

677 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

678 

1230 

1294 

1358 

1423 

1486 

1550 

1014 

1078 

1742 

1806 

64 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

64 

680 

832509 

2.Y?3 

2637 

2700 

2704 

2828 

2892 

2956 

3020 

3083 

64 

681 

3147 

321] 

3275 

3338 

3402 

3460 

3530 

3593 

3657 

3721 

64 

682 

3784 

3848 

3912 

3975 

4039 

4103 

4106 

4220 

4294 

4357 

64 

683 

4421 

4484 

4548 

4611 

4075 

4739 

4802 

4800 

4tt&' 

4993 

64 

684 

505(5 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

55(54 

5027 

C3 

685 

5091 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

63 

686 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6707 

6830 

6894 

63 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7402 

7525 

63 

688 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

689 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

838849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

63 

691 

9478 

9541 

9604 

9067 

9729 

9792 

9855 

9918 

9981 

..43 

63 

692 

"84-JlOrt 

0169 

0232 

0294 

0357 

0420 

0482 

0545 

0608 

0671 

63 

1  693 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

03 

694 

[399 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

63 

695 

toes 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

62 

696 

2009 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

62 

697 

3233 

3295 

3357 

3420 

3482 

3544 

»>0<i 

3669 

3731 

3793 

62 

,  698 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

62 

699 

4477 

4539 

4601 

4664 

4726 

4788 

•l-.-)0 

4912 

4974 

5036 

62 

N.  |   0   |   1 

2|3J4|5|6|7|8|9 

IT 

12 


A   TABLE    OP   LOGARITHMS   FROM    1    TO    10,000. 


N.  |   0   |   1   |  2    3   |  4   |   5    6     7     8  |  9    D.  • 

1  700 

845093 

5160 

52-22 

5234 

534S 

5408 

5470 

5532 

5594 

56.56 

62  ! 

701 

5718 

5780 

5842 

5904 

5966 

6023 

601)0 

6151 

6-213 

6275 

62  ! 

702 

6337 

6399 

64f51 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62  j 

703 

6955 

7017 

7079 

7141 

7202 

7284 

7326 

7383 

7449 

7511 

62 

704 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

705 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8630 

8682 

8743 

G2 

706 

8805 

8866 

8928 

8989 

9051 

9112 

9174. 

9235 

9237 

9358 

61 

707 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

850033 

0095 

0150 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

61 

709 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

11J6 

1197 

710 

851258 

1320 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

61 

711 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

713 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

714 

36d8 

3759 

3820 

3881 

3941 

4002 

4003 

4124 

4185 

4245 

61 

715 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

716 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5393 

5459 

61 

717 

5519 

5580 

5640 

5701 

57(51 

5822 

5882 

5943 

6003 

6064 

61 

;718 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

66(58 

GO 

719 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60  < 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

GO 

721 

7935 

7935 

8056 

8116 

8176 

8236 

8207 

8357 

8417 

8477 

60 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723 

9138 

9198 

9253 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

724 

9739 

9799 

9859 

9918 

9978 

-.33 

..98 

.158 

.218 

.278 

60 

725 

860338 

0393 

0453 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

727 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

728 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

•26(58 

ilii 

729 

2728 

2737 

2847 

2900 

2936 

302o 

3085 

3144 

32J4 

3:2!  S3 

60 

730 

863323 

333-2 

3442 

3501 

3501 

36-20 

3080 

3739 

3709 

3858 

59 

731 

3917 

3977 

4036 

4098 

4155 

4214 

4274 

4333 

4392 

4452 

59 

732 

4511 

4570 

4630 

4fi89 

4748 

4608 

4067 

4326 

4935 

5045 

59 

733 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

734 

5696 

5755 

5314 

5374 

5933 

5392 

G051 

6110 

6169 

6223 

59 

735 

6287 

6346 

6405 

6405 

6524 

6583 

6642 

6701 

.67(-0 

6819 

5st  - 

736 

6378 

6937 

6'J9'5 

7055 

7114 

7173 

7232 

7350 

JZ49T 

'59 

737 

7467 

7526 

7585 

7644 

7703 

77-12 

7821 

7883 

7998 

59 

738 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8527 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8938 

9056 

9114 

9173 

59 

740 

869232 

9290 

9349 

9408 

9468. 

^So25^ 

9584 

9042 

9701 

9760 

59 

741 

9818 

9877 

9935 

9994, 

^53 

•  111 

.170 

.228 

.287 

.345 

59 

742 

870404 

0462 

0521 

^9579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

743 

744 

0989 
15J3- 

104Z- 
"T631 

-«06 
1690 

1164 

1748 

1223 

1806 

1281 
1865 

1339 
1923 

1398 
1981 

1456 
2040 

1515 

2098 

58 
58 

•  745 

-"12156 

2215 

2273 

2331 

2389 

2448 

2505 

2564 

2022 

2681 

58 

746 

2739 

2797 

2355 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

58 

747 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

748 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

749 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

58 

750 

875061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

58 

751 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

G102 

6160 

58 

752 

6218 

6276 

6333 

6391 

6449 

6.507 

6564 

6622 

6680 

6737 

58 

753 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

754 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

788B 

68 

755 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8886 

8934 

8981 

9039 

57 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

758 

9669 

9726 

9784 

9841 

9898 

9956 

..13 

..70 

.127 

.185 

57 

759 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

57 

N.  |   0   |   1 

2)3(4(5    6  |   7  |   8  |  9  |  D. 

A   TABLE   OP   LOGARITHMS   PROM   1   TO    10,000. 


13 


N.  |   0     1     2    3   |   4   |  '  5    6     7  |   8  |  9    D. 

760 

880814 

0871 

0998 

0985 

1042 

1099 

U56 

1213 

1271 

1328 

57 

701 

1385 

1442 

1  HI!) 

1556 

1613 

1670 

1727 

1784 

1841 

was 

57 

762 

I'.I.V, 

201-2 

2069 

2126 

•21H3 

2240 

22!)7 

2354 

2411 

2468 

57 

703 

8583 

2581 

•2638 

2695 

•27.V2 

280!) 

9866 

21123 

2980 

3037 

57 

784 

3093 

3150 

3207 

32H4 

3391 

3377 

3434 

34!)  1 

35  18 

3605 

57 

705 

3661 

3718 

3775 

:W32 

3888 

3945 

4002 

4059 

4115 

4172 

57 

706 

4229 

4985 

4342 

4:!!)!) 

4455 

4512 

4569 

4625 

4682 

4739 

57 

167 

4795 

4859 

4909 

4!)l>5 

5032 

5078 

5135 

5192 

5248 

5305 

57 

7G8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

769 

5926 

5983 

6039 

6096 

61.5-2 

6209 

6265 

6321 

6378 

6434 

56 

770 

886491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

56 

771 

7054 

7111 

7167 

7223 

7280 

7336 

73i)2 

7449 

7505 

7561 

56 

772 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

56 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

56 

774 

8741 

8797 

8853 

8909 

8<)05 

902J 

9077 

9134 

9190 

9246 

56 

775 

9302 

9358 

9414 

9470 

9520 

9582 

9638 

9(594 

9750 

9806 

56 

776 

9862 

9918 

9974 

..30 

..86 

.141 

.197 

.253 

.309 

.365 

56 

777 

890421 

0477 

0533 

0589 

0645 

0700 

075(3 

0812 

0868 

0924 

56 

778 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

56 

779 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

56 

781 

2651 

2707 

2762 

2818 

2873 

2i)2;) 

2985 

3040 

3096 

3151 

56 

782 

3307 

3969 

3318 

3373 

3  129 

3484 

3540 

3595 

3651 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

784 

4316 

4371 

4427 

4482 

4S3& 

4593 

4648 

4704 

4759 

4814 

55 

785 

4870 

4!)-.'.-) 

4980 

5030 

5091 

5146 

5-201 

5257 

5312 

5367 

55 

78fT 

51-23 

5478 

5533 

5588 

5644 

5IKK) 

5754 

5809 

5864 

5920 

55 

787 

5975 

6030 

6140 

8195 

6251 

6306 

6:561 

6410 

6471 

55 

788 

6636 

6581 

6636 

8693 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

897627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

55 

791 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

55 

792 

8725 

8780 

8835 

8890 

8944 

891)9 

9054 

9109 

9164 

9218 

55 

793 

9273 

9:5-28 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

794 

9821 

9875 

9930 

9985 

..39 

..94 

.149 

.203 

.258 

.312 

55 

795 

900367 

0422 

0471) 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

79S 

0913 

09G6 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

797 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

793 

2003 

9057 

2112 

3166 

2-2-21 

2275 

2329 

2384 

2438 

2492 

54 

799 

2517 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54. 

800 

903093 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

:;:,•_>  i 

3578 

54 

801 

3,  i33 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

803 

4716 

4770 

4824 

4878 

4!)32 

4986 

5040 

5094 

5148 

5202 

54 

804 

5256 

5310 

5364 

5418 

5479 

5526 

5580 

5634 

5688 

5742 

54 

805 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

806 

6335 

6389 

0443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

54 

808 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8360 

8914 

8967 

54 

811 

909 

9074 

9128 

9181 

9235 

9289 

9342 

!«!Mi 

9449 

9503 

54 

812 

9556 

9610 

9303 

9716 

9770 

9823 

9877 

9930 

9984 

..37 

53 

Hi3 

SMOIliM 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

051* 

0571 

53 

814 

0!>24 

0678 

(173! 

0784 

0838 

0891 

0944 

0998 

1051 

1  104 

53 

815 

1158 

1211 

1864 

1317 

1371 

L494 

1477 

1530 

1584 

1637 

53 

,  81(5 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

53 

817 

9898 

2275 

2328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

53 

818 

2753 

9806 

2859 

•2,u:5 

2986 

3019 

3072 

3125 

3178 

3931 

53 

819 

3284 

3337 

3390 

3443 

3496 

3549 

3(302 

3655 

3708 

3761 

53 

N.  |   0 

1|2|3J4|5|G|7|8|9|D. 

A   TABLE    OP   LOGARITHMS    PROM    1    TO    10,000. 


N.    0|1     2(3     4     5    6     7  |   8  |  9    D.  1 

820 

913814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

.4237 

4290 

53  j 

8-21 

4343 

4396 

444!) 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

53 

!  «22 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

53 

823 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5709 

5822 

5875 

53 

824 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

825 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

S875 

6927 

53 

826 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

53 

827 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7825 

7978 

52 

828 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

52 

831 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

..19 

..71 

52 

832 

920123 

0176 

0228 

0280 

0332 

0384 

043G 

0489 

0541 

0593 

52 

833 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1002 

1114 

52 

834 

1166 

1218 

1270 

1322 

1374 

1420 

1478 

1530 

1582 

1634 

52 

835 

1086 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

8:J6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

52 

838 

3244 

3296 

3348 

3399 

34r>l 

3503 

3555 

3607 

3658 

3710 

52 

839 

3762 

3814 

3865 

3917 

3909 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

4331 

4383 

4434 

448« 

4538 

4589 

4641 

4693 

4744 

52 

;J  «4l 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5201 

52 

I  842 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

52 

843 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

844 

•  6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

51 

845 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7208 

7319 

51 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

51 

847 

7883 

7935 

798(5 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

51 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

51 

849 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9206 

9317 

9368 

51 

850' 

929419 

9470 

9521 

9572 

9623 

9674 

9725 

9770 

9827 

9879 

51 

851 

9930 

9981 

..32 

..83 

.134 

.185 

.236 

.287 

.338 

.389 

51 

852 

930440 

0491 

0542 

0592 

0643 

0094 

(.745 

0796 

0847 

0898 

51 

853 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1350 

1407 

51 

•854 

1458 

150!) 

1560 

1610 

16fil 

1712 

1763 

1814 

1865 

1915 

51 

855 

1966 

2017 

2008 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

850 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

51 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

859 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

51 

860 

934498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

50 

80  1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

50 

862 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

50 

863 

6011 

6061 

6111 

6162 

62J2 

6262 

6313 

6363 

6413 

6463 

50 

864 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

50 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

866 

7518 

7568 

7618 

7668 

7718 

?;()«> 

7819 

7869 

7919 

7969 

50 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

50 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

939519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

.50 

871 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

50 

872 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

874 

1511 

1501 

1611 

1000 

1710 

1760 

1809 

1859 

1909 

1958 

50 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

870 

2504 

2554 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

50 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

49 

878 

3495 

3544 

3593 

3643 

3(i92 

3742 

3791 

3841 

3890 

3939 

49 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

49 

N.    0|1|2    3j4|5|6    7  |   8  |  9  |  D. 

A    TABF.E    OP    LOGARITHMS    FROM    1    TO    10,000. 


N.     0   |   1   |  2    3   |  4     5|6|7     8    9D. 

880 

944483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

49 

881 

4976 

5025 

51174 

5134 

5173 

5222 

5272 

5321 

5370 

5419 

49 

882 

MBO 

5518 

5567 

5->16 

5(ir>,-> 

5715 

5764 

5813 

5862 

5912 

49 

883 

5901 

(5010 

6059 

6108 

8157 

6207 

6356 

6305 

G354 

6403 

49 

884 

6452 

6501 

6551 

6600 

(5649 

(56HH 

(5747 

6796 

6845 

6894 

49 

885 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7987 

7336 

7335 

49 

83(5 

7434 

7483 

7532 

7581 

7630 

7679 

77,'H 

7777 

7826 

7875 

49 

887 

7924 

7973 

H022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

888 

8413 

841)2 

8511 

8580 

8609 

8:i57 

8706 

8755 

8804 

8353 

49 

889 

8932 

8951 

8999 

9048 

9097 

111  Hi 

9195 

9244 

9292 

9341 

49 

800 

94!)39'J 

!)43!» 

9488 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

49 

81)1 

9878 

9926 

9975 

..24 

..73 

.121 

.170 

.219 

.267 

.316 

49 

893 

•JjJMi 

0414 

04(52 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

49 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49 

8i)4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1(577 

1726 

1775 

49 

895 

1823 

1872 

1920 

1969 

2017 

2:ir,ij 

2114 

2163 

2211 

2260 

48 

898 

2308 

2356 

2405 

2453 

2502 

2:,5I) 

2599 

2647 

2696 

2744 

48 

897 

9799 

2841 

288!) 

2938 

29815 

3034 

3083 

3131 

3180 

3228 

48 

898 

327t> 

3325 

3373 

3421 

3470 

3518 

35156 

3615 

3663 

3711 

48 

899 

3760 

3808 

3356 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

48 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

48 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

933 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48 

934 

6108 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

906 

71-28 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

907 

7607 

7(555 

7703 

7751 

7793 

7847 

7894 

7942 

7990 

8038 

48 

90S 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

909 

85J54 

8612 

8ti59 

8707 

8755 

88J3 

8850 

8898 

8946 

8994 

48 

910 

959041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

48 

911 

9518 

9566 

9(514 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

91-2 

9995 

..42 

..90 

.138 

.185 

.233 

.280 

.328 

.376 

.423 

48 

913 

930471 

0518 

0566 

0613 

0861 

0709 

0756 

08:)4 

0851 

0899 

48 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

915 

1421 

1469 

1516 

15(53 

1611 

1658 

1706 

1753 

1801 

1848 

47 

91(5 

1895 

1943 

19!)0 

2038 

2083 

2132 

2180 

2227 

2275 

2322 

47 

917 

2369 

2417 

2464 

2511 

2559 

21506 

2653 

2701 

2748 

2795 

47 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

919 

:wi6 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920 

953788 

3835 

3883 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

47 

923 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

47 

924 

5672 

5719 

5766 

5813 

5860 

5907 

59.54 

6001 

6048 

6095 

47 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

65(54 

47 

92f> 

6611 

6658 

6705 

6752 

6799 

(5845 

6892 

6939 

6986 

7033 

47 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7154 

7501 

47 

928 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

47 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47 

930 

968483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

47 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47 

932 

9416 

9463 

9509 

9556 

9602 

91549 

9695 

9742 

9789 

9835 

47 

1  933 

9882 

9928 

9975 

..21 

..68 

.114 

.161 

.207 

.254 

.300 

47 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

935 

0812 

(H.-H 

09!)4 

0951 

0997 

1044 

1090 

1137 

1183 

12-29 

46 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46 

937 

1740 

1786 

1832 

1879 

1925 

11171 

2018 

2064 

2110 

2157 

46 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

N.  |   0   |   1  |  2  |  3  |   4    5    6|7|8    9  |  D.  J 

16 


A    TABLE    OP    LOGARITHMS    FROM    1    TO    10.000 


N.  |   0   |   1     2    3   I  4   |   5  I  6     7|8|9|D. 

940 

973128 

3174 

3220 

32615 

3313 

3359  I  3405 

3451 

3497 

3543 

46 

941 

3590 

3!536 

3682 

3728 

3774 

3820 

38(56 

3913 

3959 

40J5 

46 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

943 

4512 

4558 

4(504 

4(550 

46% 

4742 

4788 

4S34 

4880 

4926 

46 

944 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

945 

5432 

5478 

5524 

5570 

5616 

5(5(52 

5707 

5753 

5799 

5845 

46 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

46 

917 

6350 

6396 

6442 

6488 

6533 

6579 

66-25 

6671 

6717 

6763 

46 

948 

6808 

6854 

6900 

6346 

6992 

7037 

7083 

7129 

7175 

7220 

46 

949 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

959 

977724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

46 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

81)56 

9002 

9047 

46 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

46 

954 

9548 

9594 

9639 

9;>85 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

03(57 

0412 

45 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

ms 

45 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

982271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

45 

961 

2723 

2769 

2814 

2859 

2904 

2949 

-2994 

3040 

3085 

3130 

45 

962 

3175 

3220 

32(55 

3310 

3356 

3401 

3446 

3401 

3536 

3581 

45 

983 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

964 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

9C6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426 

5471 

5516 

5561 

5606 

5051 

5696 

5741 

5786 

5830 

45 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

986772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

45 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

45 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

45 

973 

8113 

8157 

82!)2 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

45 

974 

8559 

8604 

8(548 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

1  975 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

976 

9450 

9494 

9539 

9583 

9G28 

9672 

9717 

9761 

9806 

9850 

44 

,  977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.250 

.294 

44 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

991226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

44 

981 

1669 

1713 

1758 

1802 

1848 

1890 

1935 

1979 

2023 

2067 

44 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

44 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

986 

3877 

3921 

39155 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

988 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

-5152 

44 

989 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

44 

990 

995635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

44 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6330 

6424 

64(58 

44 

992 

6512 

6555 

6599 

6643 

6!587 

6731 

6774 

6818 

f>862 

690(5 

44 

&93 

G919 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

995 

7823 

7867 

7910 

7954 

7>J98 

8041 

8085 

8129 

8172 

8216 

44 

996 

8259 

8303 

8347 

839!) 

8434 

8477 

8521 

8564 

8(508 

8652 

44 

997 

8695 

8739 

8782 

8326 

88(59 

8913 

8956 

9000 

9043 

9087 

44 

998 

9131 

9174 

9213 

»2f51 

9305 

9348 

9392 

9435 

9479 

9522 

44 

999 

9565 

9609 

9652 

9596 

973!) 

9783 

9826 

9870 

9913 

9957 

43 

N.  |   0 

1    2   |  3     4  |   5     6     7  |   8 

9   D. 

SINES  AND  TANGENTS, 


FOR   EVERY 


DEGREE   AND   MINUTE 

OF   THE   QUADRANT, 


N.B.  The  minutes  in  the  left-hand  column  of  each  page,  increasing 
downwards,  belong  to  the  degrees  at  the  top  ;  and  those  increasing  up- 
wards, in  the  right-hand  column,  belong  to  the  degrees  below. 


18 


(0  Degree.)     A  TABLE  OF  LOGARITHMIC 


M.    Sine   [   D   |   Cosine.  |  D.    Tang.  |   D    Cotang.     ! 

0 

0.000000 

10.000000 

0.000000 

Infinite. 

60 

1 

6.463726 

501717 

000000 

00 

6.4i;37-2r> 

501717 

13.536274 

59 

2 

764756 

293485 

030000 

00 

764756 

2J3433 

235-244 

58 

3 

940847 

208231 

000000 

00 

940847 

206331 

053153 

57 

4 

7.065786 

161517 

000000 

00 

7.065785 

1-J1517 

12.934214 

5<> 

5 

162695 

1319(58 

000000 

03 

162696 

1319(i9 

837304 

55 

6 

241877 

111575 

9.999993 

01 

241878 

111578 

758122 

54 

7 

:io-^24 

93653 

939393 

01 

308825 

93!553 

601175 

53 

8 

366816 

85254 

909093 

01 

366317 

85254 

633183 

52 

9 

4179S8 

76263 

999399 

01 

417970 

762(53 

532030 

51 

10 

463725 

68988 

999998 

01 

463727 

63933 

5315-273 

50 

11 

7.505118 

62981 

9.993993 

01 

7.505120 

62981 

12.494883 

49 

12 

542906 

57936 

999397 

01 

542909 

57933 

457091 

48 

13 

577668 

53641 

993997 

01 

577672 

53642 

422328 

47 

14 

609853 

49938 

999398 

01 

609857 

49039 

:uni3 

46 

15 

639816 

46714 

999395 

01 

6393-30 

46715 

3:50180 

45 

16 

667845 

43881 

993995 

01 

667849 

43832 

332151 

44 

17 

694173 

41372 

999935 

01 

694]  79 

41373 

305821 

43 

18 

718997 

39135 

999394 

01 

719J03 

39136 

230337 

42 

19 

742477 

37127 

939393 

01 

742484 

37128 

257516 

41 

20 

764754 

35315 

999933 

01 

764761 

35136 

235239 

40 

21 

7.785943 

33672 

9.999992 

01 

7.785951 

33673 

1-2.214049 

39 

22 

806146 

32175 

999991 

01 

806155 

32171) 

193845 

38 

23 

825451 

308J5 

993993 

01 

825460 

30838 

174540 

37  ! 

24 

843934 

29547 

993989 

02 

843344 

29549 

155356 

35  i 

25 

881662 

28388 

999938 

02 

881674 

28393 

138325 

35 

23 

878S95 

27317 

999383 

02 

878703 

27318 

121-2J2 

34 

27 

895085 

26323 

999937 

02 

895093 

20325 

104  9J1 

33 

28 

910879 

25399 

999985 

02 

9J0394 

25401 

083106 

32 

29 

926119 

24538 

999985 

02 

925134 

24540 

073365 

31 

30 

940842 

23733 

999983 

02 

940858 

23735 

053142 

30 

31 

7.955082 

22930 

9.9999.32 

02 

7.955100 

22981 

12.044903 

29 

32 

963870 

22273 

999381 

02 

968889 

22275 

031111 

23 

33 

982233 

21*508 

993930 

02 

932253 

21610 

017747 

27 

34 

995198 

2J331 

999379 

02 

995219 

20933 

004781 

28 

35 

8.037787 

S03UO 

933977 

02 

8.007809 

20392 

11.99-2191 

25 

38 

020021 

1U331 

9J3376 

02 

020045 

19333 

979955 

24 

37 

031919 

1930-2 

939975 

02 

031945 

19305 

9S8J55 

23 

38 

043501 

18801 

999373 

02 

043527 

18333 

956473 

22 

39    054781 

18325 

933372 

02 

054809 

18327 

945191 

21 

40 

OS5776 

17872 

993371 

02 

065806 

17874 

934194 

20 

41 

8.076500 

17441 

9.939969 

03 

8.076531 

17144 

11.9=23469 

19 

42 

088965 

17031 

999918 

02 

086397 

17034 

913303 

18 

43 

097183 

16639 

!).).).)  iii 

02 

037217 

16642 

902783 

17 

44 

107167 

16255 

9333:>4 

03 

107202 

16268 

892797 

16 

45 

116926 

15938 

993963 

03 

11(5933 

15910 

883037 

15 

46 

126471 

15566 

993931 

03 

126510 

15568 

873490 

14 

47 

135810 

15238 

939359 

03 

135851 

15241 

864149 

13 

48 

144953 

14924 

999958 

03 

144996 

149.27 

855004 

12 

49 

153907 

14522 

999356 

03 

153952 

14627 

846048 

11 

50 

162(581 

14333 

999954 

03 

162727 

14336 

837273 

10 

51 

8.171280 

14054 

9.9903.52 

03 

8.171328 

14057 

11.828672 

9 

52 

179713 

13786 

999950 

03 

179763 

13790 

820237 

8 

53 

187985 

13529 

939948 

03 

188036 

13532 

811934 

54 

193102 

13280 

993946 

03 

196156 

13284 

833844 

6 

55 

204070 

13041 

939344 

03 

204126 

115044 

795874 

5 

58- 

211895 

12810 

939342 

04 

211953 

12814 

788047 

4 

57 

219581 

12587 

993340 

04 

219541 

12590 

78J359 

3 

58 

227134 

12372 

£33338 

04 

227195 

12376 

772805 

2 

59 

234557 

12154 

993936 

04 

234621 

12168 

765379 

1 

60 

241855 

11963 

939334 

04 

241921 

11987 

758079 

0 

|  Cosine  |          Sine   |    |  Cotang.  |         Tang.   |  M. 

89  Degrees. 


SINES  AND  TANGENTS.    (1  Degree.) 


19 


M.   Sine      D.   |  Cosine  |  D.  |   Tang.  |  IX   |  Cotang. 

0 

ri.-J-HH.V) 

119U3 

9.999934 

04 

8.241921 

11967 

11.758079 

60 

1 

2-4  9033 

11768 

999932 

04 

249102 

11772 

750898 

59 

2 

256094 

11.  VO 

999929 

04 

25(1165 

11584 

743835 

58 

3 

263042 

11398 

999927 

04 

263115 

11402 

736885 

57 

4 

269881 

11221 

9!  Ml:  l-.'/i 

04 

2(>99f><; 

11225 

730044 

56  ! 

5 

276614 

11050 

9999-22 

04 

276691 

11054 

723309 

K 

1  (i 

2rf«43 

10883 

999920 

04 

283323 

10887 

716677 

54 

7 

289773 

10721 

999918 

04 

289856 

10726 

710144 

53 

8 

296807 

10565 

999915 

04 

296292 

10570 

703708 

52 

9 

302546 

10413 

99H913 

04 

3<W»i34 

10418 

697366 

51 

10 

m-o.n 

10266 

999910 

04 

308884 

10270 

691116 

50 

11 

8.314954 

10122 

9.999907 

04 

8.315046 

10126 

11.684954 

49 

12 

321027 

9982 

999905 

04 

321122 

9987 

678878 

48 

13 

327016 

9847 

999902 

04 

327114 

9851 

672886 

47 

14 

332924 

9714 

999899 

05 

333025 

9719 

666975 

46 

15 

338753 

9586 

999897 

05 

338856 

9590 

661144 

45 

16 

344504 

9460 

999894 

05 

344610 

9465 

655390 

44 

17 

350181 

9338 

999891 

05 

350289 

9343 

649711 

43 

18 

355783 

9219 

999888 

05 

355895 

9224 

644105 

42 

19 

361315 

9103 

999885 

05 

361430 

9108 

638570 

41 

20 

366777 

8990 

999882 

05 

366895 

8995 

633105 

40 

21 

8.372171 

8880 

9.999879 

05 

8.372292 

8885 

11.627708 

39 

22 

377499 

8772 

999876 

05 

377622 

8777 

622378 

38 

23 

382762 

8667 

999873 

05 

382889 

8672 

617111 

37 

24 

387962 

8564 

999870 

05 

388092 

8570 

611908 

36 

25 

393101 

8464 

999867 

05 

393234 

8470 

606766 

35 

26 

398179 

8366 

999864 

05 

398315 

8371 

601685 

34 

27 

403199 

8271 

999861 

05 

403338 

8276 

596662 

33 

28 

408161 

8177 

999858 

05 

408304 

8182 

591696 

32 

29 

413068 

8086 

999854 

05 

413213 

8091 

586787 

31 

30 

417919 

7996 

999851 

06 

418068 

8002 

581932 

30 

31 

8.422717 

7909 

9.999848 

06 

8.422869 

7914 

11.577131 

29 

32 

427462 

7823 

999844 

06 

427618 

7830 

572382 

28 

33 

432156 

7740 

999841 

06 

432315 

7745 

567685 

27 

34 

436800 

7657 

999838 

06 

436962 

7663 

563038 

26 

35 

441394 

7577 

999834 

06 

441560 

7583 

558440 

25  I 

36 

445941 

7499 

999831 

06 

446110 

7505 

553890 

24 

37 

450440 

7422 

999827 

06 

450613 

7428 

549387 

23 

38 

454893 

7346 

999823 

06 

455070 

7352 

544930 

22 

39 

459301 

7273 

999820 

06 

459481 

7279 

540519 

21 

40 

463665 

7200 

999816 

06 

4G3849 

7206 

536151 

20 

41 

8.467985 

7129 

9.999812 

06 

8.468172 

7135 

11.531828 

19 

42 

472263 

7060 

999809 

06 

472454 

7066 

527546 

18 

43 

476498 

6991 

999805 

06 

476693 

6998 

523307 

17 

44 

480693 

6924 

999801 

06 

480892 

6931 

519108 

16 

45 

484848 

6859 

999797 

07 

485050 

6865 

514950 

15 

46 

488963 

6794 

999793 

07 

489170 

6801 

510830 

14 

47 

493040 

6731 

999790 

07 

493250 

6738 

506750 

13 

48 

497078 

6669 

999786 

07 

497293 

6676 

502707 

12 

49 

501080 

6608 

999782 

07 

501298 

6615 

498702 

11 

50 

505045 

65-18 

999778 

07 

505267 

6555 

494733 

10 

51 

8.508974 

6489 

9.999774 

07 

8.509200 

6496 

11.490800 

9 

52 

512867 

6431 

999769 

07 

513098 

6439 

4<36902 

8 

53 

516726 

6375 

999765 

07 

516961 

6382 

483039 

7 

54 

520551 

6319 

999761 

07 

520790 

6326 

479210 

6 

'  55 

524343 

6264 

999757 

07 

524586 

6272 

475414 

5 

56 

528102 

6211 

999753 

07 

528349 

6218 

471651 

4 

57 

531828 

6158 

999748 

07 

532080 

6165 

467920 

3 

58 

535523 

6106 

999744 

07 

535779 

6113 

464221 

2 

59 

539186 

6055 

999740 

07 

539447 

6062 

460553 

1 

60 

542819 

6004 

999735 

07 

543084 

6012 

456916 

0 

Cosine   |       |   Sine   |    |  Cotang.  |      |   Tang.   |  M. 

89  Degrees. 


so 


(2  Degrees.)     A  TABLE  OP  LOGARITHMIC 


M.    Sine   |  D.   |  Cosine. 

D.  |   Tang.    D.   |  Cotang. 

0 

8.542819 

6004 

9.999735 

07 

8.543084 

6012 

11.456916 

60 

1 

546422 

5955 

999731 

07 

546601 

5962 

453309 

59 

2 

549995 

5906 

999726 

07 

550268 

5914 

449732 

58 

3 

553539 

5858 

999722 

08 

553817 

5866 

446183 

57 

4 

557051 

5811 

999717 

08 

557336 

5819 

442664 

56 

5 

530540 

5765 

999713 

08 

560828 

5773 

439172 

55 

.  6 

563999 

5719 

993708 

08 

564231 

57-27 

435709 

54 

7 

567431 

5674 

999704 

08 

557727 

5682 

432273 

53  ; 

8 

570836 

5630 

999699 

08 

571137 

5638 

428863 

52 

9 

574214 

5587 

999694 

08 

574520 

5595 

425480 

51 

10 

577566 

5544 

999689 

08 

577877 

5552 

422123 

50 

11 

8.580892 

5502 

9.999685 

08 

8.581208 

5510 

11.418792 

49 

12 

584193 

5460 

999G80 

08 

584514 

5468 

415486 

48 

13 

587469 

5419 

999675 

08 

587795 

5427 

412205 

47 

14 

590721 

5379 

999670 

08 

591051 

5387 

408949 

46 

15 

593948 

5339 

999665 

08 

594283 

5347 

405717 

45 

16 

597152 

5300 

999660 

08 

597492 

5308 

402508 

44 

17 

600332 

5261 

999655 

08 

600677 

5270 

399323 

43 

18 

603489 

5223 

999850 

08 

603839 

5232 

396161 

42 

19 

606623 

5186 

999645 

09 

605978 

5194 

393022 

41 

20 

609734 

5149 

999640 

09 

610094 

5158 

389906 

40 

21 

8.612823 

5112 

9.999535 

09 

8.613189 

5121 

11.386811 

39 

22 

615891 

5076 

999629 

09 

616262 

5085 

383738 

38 

23 

618937 

5041 

999624 

09 

619313 

5050 

380687 

37 

24 

621962 

5006 

999619 

09 

622343 

5015 

377657 

36 

25 

624965 

4972 

999614 

09 

625352 

4981 

374648 

35 

26 

627948 

4938 

999608 

09 

628340 

4947 

371660 

34 

27 

630911 

4904 

999603 

09 

631308 

4913 

368692 

33 

28 

633854 

4871 

999597 

09 

634256 

4880 

365744 

32 

29 

636776 

4839 

999592 

03 

637184 

4848 

362816 

31 

30 

639680 

4806 

999586 

09 

640093 

4816 

359907 

30 

31 

8.642563 

4775 

9.999581 

09 

8.642982 

4784 

11.357018 

29 

32 

645423 

4743 

999575 

09 

645853 

4753 

354147 

28 

33 

648274 

4712 

999570 

09 

648704 

4722 

351296 

27 

34 

651102 

4682 

999564 

09 

651537 

4691 

34S463 

28 

35 

653911 

4652 

999558 

10 

654352 

4661 

345648 

25 

36 

656702 

4622 

999553 

10 

657149 

4631 

342851 

24 

37 

659475 

4592 

999547 

10 

659928 

4602 

340072 

23 

38 

662230 

4563 

999541 

10 

662689 

4573 

337311 

22 

39 

664968 

4535 

999535 

10 

665433 

4544 

334567 

21 

40 

667689 

4506 

999529 

10 

668J60 

4526 

331840 

20 

41 

8.670393 

4479 

9.999524 

10 

8.670870 

4488 

11.329130 

19 

42 

673080 

4451 

999518 

10 

673563 

4461 

326437 

18 

43 

675751 

4424 

999512 

10 

676239 

4434 

323761 

17 

44 

678405 

4397 

999506 

10 

678900 

4417 

321100 

16 

45 

681043 

4370 

999500 

10 

681544 

4380 

318456 

15 

46 

683665 

4344 

999493 

10 

684172 

4354 

315828 

14 

47 

686272 

4318 

999487 

10 

686784 

4328 

313216 

13 

48 

688863 

4292 

999481 

10 

689381 

4303 

310619 

12 

49 

691438 

4267 

999475 

10 

691963 

4277 

308037 

11 

50 

693998 

4242 

999469 

10 

694529 

4252 

305471 

10 

51 

8.696543 

4217 

9.999463 

11 

8.697081 

4228 

11.302919 

9 

52 

699073 

4192 

999456 

11 

699617 

4203 

300383 

8 

53 

701589 

4168 

999450 

11 

702139 

4179 

297861 

7 

54 

704090 

4144 

999443 

11 

704646 

4155 

295354 

6 

55 

706577 

4121 

999437 

11 

707140 

4132 

292860 

5 

56 

709049 

4097 

999431 

11 

709618 

4108 

290382 

4 

57 

711507 

4074 

999424 

11 

712083 

4085 

287917 

3 

58 

713952 

4051 

999418 

11 

714534 

4062 

285465 

2 

1  59 

716383 

4029 

999411 

11 

716972 

4040 

283028 

1 

60 

718800 

4006 

999404 

11 

719396 

4017 

280604 

0 

Cosine  |      |   Sine         Cotang.  |      |   Tang. 

M 

87  Degrees. 


SINES  AND  TANGENTS.     (3  Degrees.) 


M.  |   Sine     D.   |  Cosine    D.  |  Tang,   j   D.  |   Cotang. 

0 

8.71f<HUO 

4006 

9.999404 

11 

8.719396 

4017 

ll.-JHIIiltl 

60 

1 

721204 

3984 

999398 

11 

721806 

3995 

278194 

59 

2 

723595 

.'tow 

999391 

11 

734304 

3974 

275796 

58 

3 

725972 

:w4i 

999384 

11 

3952 

273412 

57 

4 

728337 

3jn» 

999378 

11 

7-Jsiir.it 

3930 

271041 

56 

f> 

730688 

3898 

999371 

11 

731317 

3909 

268683 

55 

6 

733027 

3877 

999364 

12 

7:5:!(ii;:i 

3889 

266337 

54 

7 

735354 

3857 

999357 

12 

73.y.)!M5 

3868 

264004 

53 

8 

7:!7i;<)7 

3836 

999350 

12 

738317 

3848 

261683 

52 

9 

739969 

3816 

999343 

12 

740T.26 

3827 

259374 

51 

10 

74-J-r,!) 

3796 

W8936 

12 

7429-22 

3807 

257078 

50 

11 

8.744536 

3776 

9.999329 

12 

8.745207 

3787 

11.254793 

49 

12 

746802 

37f>li 

999322 

12 

747479 

3768 

252521 

48 

13 

7-1  !K!.-,r> 

37:17 

999315 

12 

749740 

3749 

250260 

47 

14 

751297 

3717 

999308 

12 

751989 

3729 

248011 

46 

15 

753528 

3698 

999301 

12 

754227 

3710 

245773 

45 

16 

755747 

3679 

999294 

12 

756453 

3692 

243547 

44 

17 

757955 

366  1 

999286 

12 

758668 

3673 

241332 

43 

18 

760151 

3642 

999279 

12 

760872 

3655 

239128 

42 

19 

762337 

3624 

999272 

12 

763065 

3636 

236935 

41 

20 

764511 

3606 

999265 

12 

765246 

3618 

234754 

40 

21 

8.766675 

3588 

9.999257 

12 

8.767417 

3600 

11.232583 

39 

22 

768828 

3570 

999250 

13 

769578 

3583 

230422 

38 

23 

770970 

3553 

999242 

13 

771727 

3565 

228273 

37 

24 

773101 

3535 

999235 

13 

773866 

3548 

226134 

36 

25 

775223 

3518 

999227 

13 

775995 

3531 

224005 

35 

26 

777333 

3501 

999220 

13 

778114 

3514 

221886 

34 

27 

779434 

3484 

999212 

13 

i  8{h£J2 

3497 

219778 

33 

28 

781524 

3467 

999205 

13 

782320 

3480 

217680 

32 

29 

783605 

3451 

999197 

13 

784408 

3464 

215592 

31 

30 

785675 

3431 

999189 

13 

786486 

3447 

213514 

30 

31 

8.787736 

3418 

9.999181 

13 

8.788554 

3431 

11.211446 

29 

32 

789787 

3402 

999174 

13 

790613 

3414 

209387 

28 

33 

791828 

3386 

999166 

13 

792662 

3399 

207338 

27 

34 

793859 

3370 

999158 

13 

794701 

3383 

205299 

26 

35 

795881 

3354 

999150 

13 

796731 

3368 

203269 

25 

36 

797894 

3339 

999142 

13 

798752 

3352 

201248 

24 

37 

799897 

33-23 

999134 

13 

800763 

3337 

199237 

23 

38 

801892 

3308 

999126 

13 

802765 

3322 

197235 

22 

39 

803876 

3293 

999118 

13 

804758 

3307 

195242 

21 

40 

805852 

3278 

999110 

13 

806742 

3292 

193258 

20 

41 

8.807819 

3263 

9.999102 

13 

8.808717 

3278 

11.191283 

19 

42 

809777 

3249 

999094 

14 

810683 

3262 

189317 

18 

43 

811726 

3234 

999086 

14 

812641 

3248 

187359 

17 

44 

8136G7 

3219 

9i)9077 

14 

814589 

3233 

185411 

16 

45 

815599 

3205 

999069 

14 

816529 

3219 

183471 

15 

46 

817522 

3191 

9990(51 

14 

818461 

3205 

181539 

14 

47 

819436 

3177 

999053 

14 

820384 

3191 

179616 

13 

48 

821343 

3163 

999044 

14 

822298 

3177 

177702 

12 

49 

823240 

3149 

999036 

14 

824205 

3163 

175795 

11 

50 

825130 

3135 

999027 

14 

826103 

3150 

173897 

10 

51 

8.827011 

3122 

9.999019 

14 

8.827992 

3136 

11.172008 

9 

52 

828884 

3108 

999010 

14 

829874 

3123 

170126 

8 

53 

830749 

3095 

999002 

14 

831748 

3110 

168252 

7 

54 

832607 

3082 

998993 

14 

833613 

3096 

166387 

6 

55 

834456 

3069 

998934 

14 

835471 

3083 

164529 

5 

56 

836297 

3056 

998976 

14 

837321 

3070 

162679 

4 

57 

838130 

3043 

998967 

15 

839163 

3057 

160837 

3 

68 

839956 

3030 

998958 

15 

840998 

3045 

159002 

2 

59 

841774 

3017 

998950 

15 

843835 

3032 

157175 

1 

60 

843585 

3000 

9989-11 

15 

844644 

3019 

155356 

0 

|   Cosine        |    Sine  |    |  Cotang.       |   Tang.    M  | 

86  Degrees. 


(4  Degrees.)     A  TABLE  or  LOGARITHMIC 


j  M.    Sine      D.     Cosine    IX    Tang.  |   D     Cotang.  | 

0 

8.843585 

3005 

9.998941 

15 

8.844644 

3019 

11.  155;.  6 

tk) 

1 

845387 

2992 

15 

846455 

3037 

153545 

59 

2 

847183 

2980 

996923 

15 

848-260 

2995 

1517  10 

",-: 

3 

848971 

25)07 

998914 

15 

850057 

2982 

149943 

57 

4 

850751 

2955 

998905 

15 

851846 

2370 

148154 

53 

5 

85-25-25 

2943 

998896 

15 

853623 

2958 

14!>:i72 

55 

6 

854231 

2931 

998387 

15 

855403 

2946 

144597 

54 

7 

856049 

2919 

998378 

15 

857171 

2095 

142329 

53  ! 

8 

8578J1 

291)7 

998339 

15 

858932 

3833 

1410(53 

52  1 

9 

859546 

23i)5 

933860 

15 

830683 

2911 

133314 

51 

10 

861283 

2884 

993351 

15 

,  832433 

9M 

137567 

50 

11 

8.81)3014 

2373 

9.993841 

15 

8.864173 

2888 

11,135827 

49 

12 

86473S 

28(>1 

993332 

15 

835906 

2377 

134094 

48 

13 

836455 

2850 

993323 

16 

807632 

2366 

1323(38 

47 

14 

838165 

2339 

993813 

16 

839351 

2354 

130648 

46 

15 

8693158 

2828 

998804 

16 

871064 

2343 

128336 

45 

16 

871565 

2817 

993795 

16 

872770 

2W32 

127230 

44 

17 

873255 

2306 

998785 

16 

874469 

2321 

125531 

43 

18 

874938 

2795 

998776 

16 

876162 

2811 

123333 

42 

19 

876615 

2783 

993766 

16 

877849 

2800 

122151 

41 

20 

878285 

2773 

993757 

16 

879529 

2789 

120471 

40 

21 

8.879949 

2763 

9.998747 

16 

8.831202 

2779 

11.118793 

39 

22 

881607 

2752 

998738 

16 

832369 

2768 

117131 

38 

23 

88-3258 

2742 

998728 

16 

884.530 

2758 

115470  • 

37 

24 

834903 

2731 

998718 

16 

836185 

2747 

113815 

36 

25 

886542 

2721 

998708 

16 

887833 

2737 

112167 

35 

26 

888174 

2711 

998:593 

16 

889476 

2727 

110524 

34 

27 

889801 

2700 

998689 

16 

891112 

2717 

108888 

33 

28 

891421 

2693 

993679 

16 

83274-2 

2707 

107258 

32 

29 

893035 

2680 

998G69 

17 

894366 

2697 

1056-34 

31 

30 

894643 

2870 

998659 

17 

895984 

2687 

101016 

30 

3] 

8.896246 

2660 

9.993649 

17 

8.897593 

2677 

11.102404 

29 

32 

897842 

2651 

05)3639 

17 

839203 

2667 

100797 

28 

33 

89943-2 

2641 

983029 

17 

90J803 

2658 

099197 

27 

34 

901017 

3f!31 

9931519 

17 

902398 

2648 

097602 

26 

35 

90-259-5 

21522 

9j3iit!9 

17 

903937 

2338 

093:)13 

25 

36 

904169 

2612 

903593 

17 

905570 

2629 

09-1430 

24 

37 

90573(5 

2(503 

938539 

17 

9J7147 

2620 

032353 

23 

38 

907-297 

2593 

933578 

17 

908719 

2610 

091281 

22 

39 

908353 

2584 

998568 

17 

910235 

2301 

089715 

21 

40 

910104 

2575 

938558 

17 

911846 

2592 

088154 

20 

41 

8.911949 

2566 

9.998548 

17 

8.913401 

2583 

11.086599 

19 

42 

913483 

2556 

993537 

17 

914951 

2574 

085049 

18 

43 

915022 

2547 

933527 

17 

913495 

2565 

083505 

17 

44 

916550 

2538 

998516 

18 

913034 

2556 

081966 

16 

45 

918073 

2529 

9i>8.503 

18 

919563 

2547 

080432 

15 

46 

919591 

2520 

998495 

18 

921096 

2538 

078904 

14 

47 

921103 

2512 

998485 

18 

922619 

2530 

077381 

13 

48 

922610 

2503 

998474 

18 

924136 

2521 

075864 

12 

49 

924112 

2494 

998461 

18 

9-25649 

2512 

074351 

11 

50 

925809 

2486 

998453 

18 

9-27156 

2503 

072344 

10 

51 

8.927100 

2477 

9.998442 

18 

8.928658 

2495 

11.071342 

9 

52 

9-28587 

2469 

993431 

18 

930155 

2486 

069345 

8 

53 

930068 

2460 

998421 

18 

931647 

2478 

068353 

54 

931544 

2452 

933410 

18 

933134 

2470 

0(508(56 

6 

55 

933015 

2443 

993399 

18 

934(516 

2461 

0(55384 

5 

56 

934481 

2435 

933333 

18 

936093 

2453 

033907 

4 

57 

93594-2 

2427 

993377 

18 

937565 

2445 

062435 

3 

58 

937398 

2419 

998366 

18 

939032 

2437 

060968 

o 

59 

9383.50 

2411 

998355 

18 

940434 

2430 

059506 

1 

60 

940-295 

2403 

938344 

18 

941952 

2421 

053048 

0 

Cosine        |   Sine   | 

Cotang.  1      |   Tang.    M. 

85  Degrees. 


SINKS  AND  TANGENTS.     (5  Degrees.) 


M.    Sine  |   D.  |  Cosine  |  D.    Tang.  |   D.    Cotang.  | 

0 

8.940296 

2403 

9.91K544 

19 

8.94111.12 

2421 

H.t).iS,|.lH 

60 

1 

941738 

2394 

938333 

19 

943404 

2413 

056596 

59 

2 

943174 

21587 

1)118322 

19 

9448.i2 

2405 

0551-18 

58 

3 

944606 

2379 

99831] 

19 

946295 

2397 

053705 

57 

4 

946034 

2371 

998300 

19 

947734 

2390 

(1.122116 

56 

5 

9474,-)ti 

3363 

998289 

1J 

949  Hi8 

2382 

050832 

55 

6 

948874 

23.-).-) 

998277 

19 

950597 

2374 

049-103 

54 

7 

950287 

998266 

19 

953031 

2366 

047979 

53 

8 

95169!) 

2340 

9  i  J255 

19 

953441 

2360 

046559 

52 

9 

953100 

3332 

'..;>  Ji:-. 

19 

954856 

2351 

045144 

51  i 

10 

954499 

HM 

993232 

19 

956267 

2344 

043733 

50 

11 

8.955894 

2317 

9,999830 

19 

8.957674 

2337 

11.042326 

49 

12 

957384 

2310 

998209 

19 

959075 

23tt 

040925 

48 

13 

958670 

23,12 

998197 

19 

960473 

2323 

039527 

47 

14 

960052 

UBS 

998186 

19 

981856 

2314 

038134 

46 

15 

961429 

2288 

998174 

J9 

9;;32.15 

2307 

036745 

45 

16 

962801 

22.80 

998163 

19 

984639 

2300 

035361 

44 

17 

954170 

2273 

998151 

19 

986019 

2293 

033981 

43 

13 

965534 

2266 

91)8139 

20 

967394 

2286 

032605 

42 

19 

966893 

2-2.".:  > 

998128 

20 

968766 

2279 

031234 

41 

20 

908349 

2359 

998116 

20 

970133 

2271 

029867 

40 

21 

8.969(500 

2214 

9.998104 

21) 

8.971495 

22G5 

11.028504 

39 

22 

970947 

2238 

998092 

20 

972855 

2257 

027145 

38 

23 

972289 

2231 

998080 

20 

974209 

2251 

025791 

37 

24 

973628 

2224 

993068 

20 

975560 

2244 

024440 

36 

25 

974962 

2217 

'I9S1.H; 

20 

976906 

2237 

023094 

35 

26 

976293 

2210 

998044 

20 

978248 

2230 

021752 

34 

27 

977619 

2303 

998032 

20 

979586 

2223 

020414 

33 

28 

978941 

2197 

998020 

20 

98093] 

2217 

019079 

32 

29 

980259 

2190 

998008 

20 

932251 

2-210 

017749 

31 

30 

981573 

2183 

997998 

20 

983577 

2204 

016423 

30 

31 

8.982883 

2177 

9.997984 

20 

8.984899 

21-97 

11.015101 

29 

32 

934189 

2170 

997972 

2J 

986217 

2191 

013783 

28 

33 

935191 

2163 

997959 

20 

987532 

2184 

012468 

27 

34 

986789 

2157 

997917 

20 

938342 

2178 

011158 

26 

35 

988083 

2150 

997935 

31 

990149 

2171 

009351 

25 

36 

98J374 

2144 

997922 

21 

991451 

2165 

008549 

24 

37 

99.1660 

2138 

997910 

21 

992750 

2158 

OJ7250 

23 

38 

991943 

2131 

9J78U7 

31 

994045 

2152 

005955 

22 

39 

9:13222 

2125 

997885 

21 

995337 

2146 

004(5(53 

21 

40 

994497 

2119 

997872 

21 

9-J6624 

2140 

003376 

20 

41 

8.9D5708 

2112 

9.997850 

21 

8.997908 

2134 

11.002092 

19 

42 

99703fi 

2106 

997847 

21 

999188 

2127 

000812 

18 

43 

998291) 

2100 

997835 

21 

9.000465 

2121 

10.999535 

17 

44 

999560 

2094 

997822 

21 

001733 

2115 

993262 

16 

45 

9.000816 

2087 

997809 

21 

003007 

2109 

996993 

15 

46 

002069 

2082 

997797 

21 

004272 

2103 

995728 

14 

47 

003313 

2076 

997784 

21 

005534 

2097 

994466 

13 

48 

0045(53 

2070 

997771 

21 

006792 

2091 

993208 

12 

49 

005805 

2064 

997758 

21 

008047 

2035 

991953 

11 

50 

007044 

2058 

997745 

21 

009298 

2080 

990702 

10 

51 

9.008278 

2052 

9.997732 

21 

9.010546 

2074 

10.989454 

9 

52 

009510 

2046 

997719 

21 

011790 

2068 

988210 

8 

53 

010737 

2040 

997706 

21 

013031 

2062 

986959 

7 

54 

01  1962 

2034 

997693 

22 

014268 

2056 

985732 

6 

55 

013182 

2029 

997680 

22 

015502 

2051 

984498 

5 

56 

014400 

2023 

9976(57 

22 

(11  (57:52 

2045 

983268 

4 

57 

015613 

2017 

997654 

22 

017959 

2040 

982041 

3 

58 

016824 

2012 

997041 

22 

019183 

2033 

980817 

2 

59 

(50 

018031 
019235 

2006 
2000 

997628 
997614 

22 
22 

020403 
021620 

2028 
2023 

979597 
978380 

1 
0 

|  Cosine  |      |   Sine   |     Cotang.  |      |    Tang.   M.  | 

84  Degress. 


(6  Degrees.)    A  TABLE  OF  LOGARITHMIC 


M.  \   Sine  |   D.    Cosine   |  D.  |  Tang.     D.     Cotang. 

0 

9.019235 

2000 

9.997614 

22 

9.021620 

2023 

10.978380 

60 

1 

020435 

1995 

997601 

22 

022834 

2017 

977166 

59 

2 

021632 

1989 

997588 

22 

024044 

2011 

975956 

58 

3 

022825 

1984 

997574 

22 

025-251 

2006 

974749 

57 

4 

024016 

1978 

997561 

23 

026455 

2000 

973545 

56 

5 

.  025203 

1973 

997547 

22 

027655 

1995 

972345 

55 

6 

026386 

1967 

997534 

23 

028852 

1990 

971148 

54 

7 

0275S7 

1962 

997520 

23 

030046 

1985 

969954 

53 

8 

028744 

1957 

997507 

23 

031237 

1979 

968763 

52 

9 

029918 

1951 

997493 

23 

032425 

1974 

967575 

5]  , 

M 

031089 

1947 

997480 

23 

033^09 

1969 

966391 

50 

11 

9.032257 

1941 

9.997466 

23 

9.034791 

1964 

10.965209 

49 

12 

033421 

1936 

997452 

23 

035969 

1958 

964031 

48 

13 

034582 

1930 

997439 

23 

037144 

1953 

962856 

47 

14 

035741 

1925 

997425 

23 

038316 

1948 

961684 

46 

15 

036896 

1920 

997411 

23 

039485 

1943 

960515 

45 

16 

038048 

1915 

997397 

23 

040651 

1938 

959349 

44 

17 

039197 

1910 

997383 

23 

041813 

1933 

958187 

43 

18 

040342 

1905 

997369 

23 

042973 

1928 

957027 

42 

19 

041485 

1899 

997355 

23 

044130 

1923 

955870 

41 

20 

042625 

1894 

997341 

23 

045284 

1918 

9547U6 

40 

21 

9.043762 

1889 

9.997327 

24 

9.046434 

1913 

10.953566 

39 

22 

044895 

1884 

997313 

24 

047582 

1908 

952418 

38 

23 

046026 

1879 

997299 

£1 

048727 

1903 

951273 

37 

24 

047154 

1875 

997285 

24 

049869 

1898 

950131 

36 

25 

048279 

1870 

997271 

24 

051008 

1893 

948992 

35 

26 

049400 

1865 

997257 

24 

052144 

1889 

947856 

34 

27 

050519 

1860 

997242 

24 

053277 

1884 

946723 

33 

28 

051635 

1855 

997228 

24 

054407 

1879 

945593 

32 

29 

052749 

1850 

997214 

24 

055535 

1874 

944465 

31 

30 

053859 

1845 

997199 

24 

056659 

1870 

943341 

30 

31 

9.054966 

1841 

9.997185 

24 

9.057781 

1865 

10.942219 

29 

32 

056071 

1836 

997170 

24 

058900 

1869 

941100 

28 

33 

057172 

1831 

997156 

24 

060016 

1855 

939984 

27 

34 

058271 

1827 

997141 

24 

061130 

1851 

938870 

26 

35 

059367 

1822 

997127 

24 

062240 

1846 

937760 

25 

36 

0604(50 

1817 

997112 

24 

063348 

1842 

936652 

24 

37 

"  061551 

1813 

997098 

24 

064453 

1837 

935547 

23 

38 

OG2639 

1808 

997083 

25 

OC.5556 

1833 

934444 

22 

39 

063724 

1804 

997068 

25 

066655 

1828 

933345 

al- 

40 

064806 

1799 

997053 

25 

067752 

1824 

932248 

so 

41 

9.065885 

1794 

9.997039 

25 

9.068^41} 

1819 

10.931154 

19 

42 

066902 

1790 

997024 

25 

069938 

1815 

930062 

18 

43 

068036 

1786 

997009 

25 

071027 

1810 

928973 

17 

44 

069107 

1781 

996994 

25 

072113 

1806 

927887 

16 

45 

070176 

1777 

996979 

25 

073197 

1802 

926803 

15 

46 

071242 

1772 

996904 

25 

074278 

1797 

925722 

14 

47 

072306 

1768 

996949 

25 

075356 

1703 

924644 

13 

48 

073366 

1763 

996934 

25 

076432 

1789 

923568 

12 

49 

074424 

1759 

995919 

25 

077505 

1784 

922495 

1] 

50 

075480 

1755 

996904 

25 

078576 

1780 

921424 

10 

51 

9.076533 

1750 

9.996889 

25 

9.079G44 

1776 

10.920356 

9 

52 

077583 

1746 

996874 

25 

080710 

1772 

919290 

8 

53 

078631 

1742 

996858 

25 

081773 

1767 

918227 

7 

54 

879676 

1738 

996843 

25 

082833 

1763 

917167 

6 

55 

080719 

1733 

996828 

25 

083891 

1759 

916109 

5 

56 

081759 

1729 

996812 

26 

084947 

1755 

915053 

4 

57 

082797 

1725 

996797 

26 

086000 

1751 

914000 

3 

58 

0&3832 

1721 

996782 

26 

087050 

1747 

912950 

2 

59 

084804 

1717 

996766 

26 

Q88098 

1743 

911902 

1 

60 

085894 

1713 

996751 

26 

089144 

1738 

910856 

0 

|   Cosine  |      |   Sine   |      Cotang.        |   Tang.    M. 

83  Degrees. 


SINES  AND  TANGENTS.     (7  Degrees.) 


M.  |   Sine   |   D.  |  Cosine  |  D.    Tang.     D.  |  Cotang.  | 

0 

9.085894 

1713 

9.996751 

26  1  9.089144 

1738 

10.910856 

60 

1 

086922 

170!) 

998735 

26  |   090187 

1734 

909813 

59 

2 

067947 

1704 

<»<)07-20 

20 

091228 

1730 

908772 

58 

3 

088970 

1700 

998764 

2(5 

092206 

17-27 

907734 

57 

4 

089990 

1696 

990088 

26 

093302 

1722 

906698 

56 

5 

091008 

1692 

996673 

26 

094336 

1719 

905664 

55 

6 

099024 

1688 

!i:tOO:>7 

26 

095367 

1715 

904633 

54 

7 

093037 

1684 

9915041 

26 

090395 

1711 

903605 

53 

8 

09-1047 

1680 

9K0025 

26 

097422 

1707 

902578 

52 

9 

<)'j.->nr>r> 

1676 

998610 

26 

098446 

1703 

901554 

51 

10 

OSMiOTrt 

1673 

996594 

26 

099468 

1699 

900532 

50 

11 

9.097005 

1668 

9.996578 

27 

9.100487 

1695 

10.899513 

49 

K 

008866 

1605 

990562 

27 

101504 

1091 

898496 

48 

13 

099005 

1(561 

99(5546 

27 

102519 

1687 

897481 

47 

14 

100002 

1657 

996530 

27 

103532 

1084 

896468 

46 

i  ir> 

10105G 

1653 

996514 

27 

104542 

1680 

895458 

45 

16 

1H2048 

1649 

996498 

27 

105550 

1676 

894450 

44 

17 

103037 

1645 

996482 

2? 

106556 

1672 

893444 

43 

18 

104025 

1641 

996465 

O1? 

107559 

1669 

892441 

42 

19 

iu.-,oio 

1638 

996449 

27 

108560 

1665 

891440 

41 

20 

105992 

1634 

996433 

27 

109559 

1661 

890441 

40 

21 

9.100973 

1630 

9.996417 

27 

9.110556 

1658 

10.889444 

39 

33 

107951 

1627 

996400 

27 

111551 

1654 

888449 

38 

33 

108927 

1623 

996384 

27 

112543 

1650 

887457 

37 

24 

109901 

1619 

996368 

27 

113533 

1646 

886467 

36 

25 

1  10--73 

1616 

996351 

27 

114521 

1643 

885479 

35 

2(5 

111842 

1612 

990335 

27 

115507 

1639 

884493 

34 

27 

112809 

1608 

996318 

27 

110491 

1636 

883509 

33 

28 

113774 

1605 

996302 

28 

117472 

1632 

882528 

32 

29 

114737 

1601 

996385 

28 

118452 

1629 

881548 

31 

10 

115698 

1597 

996269 

28 

119429 

1625 

880571 

30 

31 

9.116650 

1594 

9.998252 

28 

9.120404 

1622 

10.879596 

29 

32 

117613 

1590 

996235 

28 

121377 

1618 

878623 

28 

83 

118567 

1587 

996219 

28 

122348 

1615 

877652 

27 

34 

119519 

1583 

990202 

28 

123317 

1611 

876683 

26 

35 

12040!) 

1580 

996185 

23 

124284 

1607 

875716 

25 

36 

121417 

1570 

996168 

28 

125249 

1604 

874751 

24 

37 

12-2302 

1573 

990151 

38 

126211 

1601 

873789 

23 

38 

1233015 

1569 

9915134 

28 

127172 

1597 

872828 

22 

39 

124248 

1306 

990117 

28 

128130 

1594 

871870 

21 

40 

125187 

1562 

996100 

28 

129087 

1591 

870913 

20 

41 

9.12(5125 

1559 

9.99(5083 

29 

9.130041 

1587 

10.869959 

19 

42 

127060 

1556 

996066 

29 

130994 

1584 

869006 

18 

43 

127993 

1552 

996049 

29 

131944 

1581 

868056 

17 

44 

1*8925 

1549 

99(5032 

29 

13-W93 

1577 

867107 

16 

45 

129854 

1545 

99(5015 

29 

133839 

1574 

800101 

15 

40 

130781 

1542 

995998 

29 

134784 

1571 

865216 

14 

47 

131706 

1539 

995980 

29 

135726 

1567 

864274 

13 

48 

132030 

1535 

995963 

29 

136667 

1564 

863333 

12 

49 

133551 

1532 

995946 

29 

137605 

1561 

862395 

11 

50 

134470 

1529 

995928 

29 

138542 

1558 

861458 

10 

51 

9.135337 

1525 

9.995911 

29 

9.139476 

1555 

10.860524 

9 

52 

138303 

1522 

995894 

29 

140409 

1551 

859591 

8 

53 

137216 

1519 

995876 

29 

141340 

1548 

£58660 

7 

54 

138128 

1516 

995859 

29 

142269 

1545 

857731 

6 

55 

139037 

1512 

995841 

29 

143196 

1542 

856804 

5 

50 

139914 

1509 

995823 

29 

144121 

1539 

855879 

4 

57 

140850 

150(5 

995806 

29 

145044 

1535 

854956 

3 

58 

11  17.V, 

1503 

995788 

29 

145966 

1532 

854034 

2 

59 

14-20f>:> 

1500 

91)5771 

29 

14(5885 

1529 

853115 

1 

60 

143555 

1496 

995753 

29 

1  17.-03 

1526 

852197 

0 

'    |  Cosine  |       |   Sine   |    |  Cotang.  |      |    Tang.  |  M.  1 

82  Degrees. 

2 

(8  Degrees.)    A  TABLE  OF  LOGARITHMIC 


M.  |   Sine   |   D   |   Cosine   |  D.  |   Tang.  |   D.   |  Cotang.  | 

0 

9.143555 

1496 

9.995753 

30 

9.147803 

1526 

10.852197 

(iO 

1 

144453 

1493 

995735 

30 

148718 

1523 

851282 

59 

2 

145349 

1490 

995717 

30 

149832 

1520 

8503(38 

58 

3 

146243 

1487 

995699 

30 

150544 

1517 

849456 

57 

4 

147136 

1484 

995681 

30 

151454 

1514 

848546 

50 

5 

148026 

1481 

995664 

30 

152363 

1511 

847637 

55 

6 

148915 

1478 

995646 

30 

153269 

1508 

846731 

54 

7 

149802 

1475 

995628 

30 

154174 

1505 

845826 

53 

8 

150686 

1472 

995610 

30 

155077 

1502 

844923 

52 

9 

151569 

1469 

995591 

30 

155978 

1499 

844022 

51 

10 

152451 

1466 

995573 

30 

156877 

1496 

843123 

50 

11 

9.153330 

1463 

9.995555 

30 

9.157775 

1493 

10.842225 

49 

12 

154208 

1460 

995537 

30 

158671 

1490 

841329 

48 

13 

155083 

1457 

995519 

30 

159565 

1487 

840435 

47 

14 

155957 

1454 

995501 

31 

160457 

1484 

839543 

46 

15 

156830 

1451 

995482 

31 

161347 

1481 

838C53 

45 

16 

157700 

1448 

995464 

31 

162236 

1479 

837764 

44 

17 

158569 

1445 

995446 

31 

163123 

1476 

836877 

43 

18 

159435 

1442 

995427 

31 

164008 

1473 

835992 

42 

19 

160301 

1439 

995409 

31 

164892 

1470 

835108 

41 

20 

161164 

1436 

995390 

31 

165774 

1467 

834226 

40 

21 

9.162025 

1433 

9.995372 

31 

9.166654 

1464 

10.833346 

39 

22 

162885 

1430 

995353 

31 

167532 

1461 

832468 

38 

23 

163743 

1427 

995334 

31 

168409 

1458 

831591 

37 

24 

164(100 

1424 

995316 

31 

169284 

1455 

8307]  6 

36 

25 

165454 

1422 

995297 

31 

170157 

1453 

829843 

35 

26 

166307 

14J9 

995278 

31 

171029 

1450 

828971 

34 

27 

167159 

1416 

9952CO 

31 

.171899 

1447 

828101 

33 

28 

168008 

1413 

995241 

32 

172767 

1444 

827233 

32 

29 

168856 

1410 

995222 

32 

173634 

1442 

826366 

31 

30 

169702 

1407 

995203 

32 

174499 

1439 

825501 

30 

31 

9.170547 

1405 

9.995184 

32 

9.175362 

1436 

10.824638 

29 

32 

171389 

1402 

995165 

32 

176224 

1433 

823776 

28 

33 

172230 

1399 

995146 

32 

177084 

1431 

822916 

27 

34 

173070 

1396 

995127 

32 

177942 

1428 

822058 

26 

35 

173908 

1394 

995108 

32 

178799 

1425 

821201 

25 

36 

174744 

1391 

995089 

32 

179655 

1423 

820345 

24 

37 

175578 

1388 

995070 

32 

180508 

1420 

819492 

23 

38 

176411 

1386 

995051 

32 

181360 

1417 

818640 

22 

39 

177242 

1383 

995032 

32 

182211 

1415 

817789 

21 

40 

178072 

1380 

995013 

32 

183059 

1412 

816941 

20 

41 

9.178900 

1377 

9.994993 

32 

9.183907 

1409 

10.816093 

19 

42 

179726 

1374 

994974 

32 

184752 

1407 

815248 

18 

43 

180551 

1372 

994955 

32 

185597 

1404 

814403 

17 

44 

181374 

1369 

994935 

32 

186439 

1402 

8]  3561 

16 

45 

182196 

1366 

994916 

33 

187280 

1399 

812720 

15 

46 

183016 

1364 

994896 

33 

188120 

1396 

811880 

14 

47 

183834 

1361 

994877 

33 

188958 

13!)3 

811042 

13 

48 

184651 

1359 

994857 

33 

189794 

1391 

810206 

12 

49 

185466 

1356 

994838 

33 

190629 

1389 

809371 

11 

50 

186280 

1353 

994818 

33 

191462 

1386 

808538 

10 

51 

9.187092 

1351 

9.994798 

33 

9.192294 

1384 

10.807706 

9 

52 

187903 

1348 

994779 

33 

193124 

1381 

806876 

8 

53 

188712 

1346 

994759 

33 

193953 

1379 

806047 

7 

54 

189519 

1343 

994739 

33 

194780 

1376 

805220 

6 

55 

190325 

1341 

994719 

33 

195606 

1374 

804394 

5 

56 

191130 

1338 

994700 

33 

196430 

1371 

803570 

4 

57 

191933 

1336 

994680 

33 

197253 

1369 

802747 

3 

58 

192734 

1333 

994660 

33 

198074 

1366 

801926 

2 

59 

193534 

1330 

994640 

33 

198894 

1364 

801106 

1 

60 

194332 

1328 

994020 

33 

199713 

1361 

800287 

0 

|  Cosine  |      |    Sine   |     Cotang.  |       |   Tang.  |  M.  ) 

81  Degrees. 


SINES  AND  TANGENTS.     (9  Degrees.) 


27 


M.  |   Sine      D.     Cosine  |  D.  |   Tang.     D.   |  Cotang.  | 

0 

9.191332 

1338 

9.1)94620 

33 

9.  1!U?13 

13til 

10.800287 

60 

1 

1951-29 

13-26 

994000 

33 

2005-29 

1359 

799471 

59 

2 

195935 

1333 

994580 

33 

201345 

1356 

798655 

58 

3 

196719 

1321 

!):)4500 

34 

202159 

1354 

797841 

57 

4 

W7511 

1318 

994.-)  10 

34 

202971 

1352 

797038 

56 

5 

198303 

13  10 

994519 

34 

203782 

1349 

796218 

55 

6 

199091 

1313 

994499 

34 

204592 

1347 

795408 

54 

7 

199879 

1311 

994479 

34 

205400 

1345 

794600 

53 

8 

900666 

1308 

991459 

34 

200207 

1342 

793793 

52 

9 

"2()]4r>i 

1306 

994438 

34 

207013 

1340 

792987 

51 

10 

202234 

1304 

994418 

34 

207817 

1338 

792183 

50 

11 

9.203017 

1301 

9.934397 

34 

9.208619 

1335 

10.791381 

49 

19 

203797 

1299 

994377 

34 

209420 

1333 

790.580 

48 

13 

204577 

l-2i)li 

994357 

34 

210220 

1331 

789780 

47 

14 

205354 

1294 

994330 

34 

211018 

1328 

788982 

46 

15 

2l)!>131 

1292 

994316 

34 

211815 

1326 

788185 

45 

16 

2(«G906 

1289 

994295 

34 

212611 

1324 

787389 

44 

17 

207679 

1287 

994274 

35 

213405 

1321 

786595 

43 

18 

208452 

1285 

994254 

35 

214198 

1319 

785802 

42 

19 

209222 

1282 

991233 

35 

214989 

1317 

785011 

41 

20 

20D9J2 

1280 

994212 

35 

215780 

1315 

784220 

40 

21 

9.210760 

1278 

9.994191 

35 

9.216568 

1312 

10.783432 

39 

22 

211526 

127fr 

994171 

35 

217356 

1310 

782644 

38 

23 

212291 

1273 

994150 

35 

218142 

1308 

781858 

37 

24 

213055 

1271 

994129 

35 

218926 

1305 

781074 

36 

25 

213818 

1268 

91)4108 

35 

219710 

1303 

780290 

35 

26 

214579 

1286 

991087 

35 

2204P2 

1301 

779508 

34 

27 

215338 

1264 

994086 

35 

221272 

1299 

778728 

33 

28 

216097 

1261 

994045 

35 

222052 

1297 

777948 

32 

29 

216854 

1259 

994024 

35 

222830 

1294 

777170 

31 

30 

217609 

1257 

994003 

35 

223606 

1292 

776394 

30 

31 

9.218333 

1255 

9.993981 

35 

9.224382 

1290 

10.775618 

29 

32 

219116 

1253 

993960 

35 

225156 

1288 

774844 

28 

33 

219868 

1250 

993939 

35 

225929 

1286 

774071 

27 

34 

220618 

1248 

993918 

35 

226700 

1284 

773300 

26 

35 

221367 

1246 

99381)6 

36 

227471 

1281 

772529 

25 

36 

222115 

1244 

993875 

36 

228239 

1279 

771761 

24 

37 

222861 

1242 

993854 

36 

229007 

1277 

770993 

23 

38 

223606 

1239 

993832' 

36 

229773 

1275 

770227 

22 

39 

224349 

1237 

993811 

36 

230539 

1273 

769461 

21 

40 

225092 

1235 

993789 

36 

231302 

1271 

768698 

20 

41 

9.225833 

1233 

9.993768 

36 

9.232065 

1269 

10.767935 

19 

42 

226573 

1231 

993746 

36 

232826 

1267 

767174 

18 

43 

227311 

1228 

993725 

36 

233586 

1265 

701)414 

17 

44 

228048 

12-26 

993703 

36 

234345 

1262 

765655 

16 

45 

223784 

1224 

993681 

36 

235103 

1260 

764897 

15 

46 

229518 

1222 

993660 

36 

235859 

1258 

764141 

14 

47 

230252 

1220 

993638 

36 

236614 

1256 

763380 

13 

48 

230984 

1218 

993616 

36 

237368 

1254 

762632 

12 

49 

231714 

1216 

9J3594 

37 

238120 

1252 

761880 

11 

50 

232444 

1214 

993572 

37 

238872 

1250 

761128 

10 

51 

9.233172 

1212 

9.993550 

37 

9.239022 

1248 

10.760378 

9 

52 

233899 

1209 

993528 

37 

240371 

1246 

759629 

8 

53 

234625 

1207 

993506 

37 

241118 

1244 

758882 

7 

54 

235349 

1205 

993484 

37 

241865 

1242 

758135 

6 

55 

23S073 

12!)3 

993402 

37 

342!)  10 

1240 

757390 

5 

56 

23(5795 

1201 

993440 

37 

243354 

1238 

750646 

4 

57 

237515 

1199 

993418 

37 

244097 

1236 

755903 

3 

58 

238235 

1197 

993390 

37 

244839 

1234 

755161 

2 

59 

238953 

1195 

993374 

37 

245579 

1232 

754421 

1 

60 

239570 

1193 

993351 

37 

246319 

1230 

753681 

0 

|  Cosine        |   Sine  |    |  Cotang.           Tang.  |  M. 

85  Degrees. 


(10  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.  j   Sine     D.    Cosine    D.  |  Tang.     D.     Cotang. 

0 

9.239670 

1193 

9.993351 

37 

9.246319 

1-230   10.753681 

60 

1 

24033G 

1191 

993329 

37 

217.157 

1228 

752943 

59 

2 

241101 

1189 

993307 

37 

247794 

1226 

752206 

58 

3 

211814 

1187 

9932-55 

37 

248530 

1224 

751470 

57 

4 

342526 

1185 

9932IJ2 

37 

219-254 

1-222 

750736 

56 

5 

2432  57 

1183 

993240 

37 

249998 

122:) 

750002 

55 

6 

243947 

1181 

993-217 

38 

259730 

1218 

74!(270 

54 

7 

244656 

1179 

993195 

38 

251461 

1217 

748539 

53 

8 

2453(53 

1177 

99317-2 

38 

252191 

1215 

747809 

52 

9 

246009 

1175 

993149 

38 

25-29-20 

1213 

747080 

51 

10 

246775 

1173 

993127 

38 

253648 

1211 

74035-2 

50 

11 

9.247478 

1171 

9.993104 

38 

9.254374 

1209 

10.745f>2i) 

49 

12 

248181 

1169 

9930H1 

38 

255100 

1207 

744900 

48 

13 

248883 

1167 

993059 

38 

255824 

1205 

744176 

47 

14 

249583 

1165 

993036 

38 

256547 

1203 

743453 

46 

15 

25028-2 

1163 

993013 

38 

257269 

1201 

742731 

45 

16 

250980 

1161 

992990 

38 

257990 

1200 

742010 

44 

17 

251677 

1159 

99-2957 

38 

258710 

1198 

741290 

43 

18 

252373 

1153 

'.u-jii  11 

38 

2594-29 

1196 

740571 

42 

19 

253067 

1156 

9929-21 

38 

260146 

1194 

73>»8.)4 

41 

20 

253761 

1154 

992898 

38 

260863 

1192 

739137 

40 

21 

9.254453 

1152 

9.992375 

38 

9.261578 

1190 

10.738422 

39 

22 

255144 

1150 

992852 

38 

25-229-2 

1189 

737703 

38 

23 

255834 

1148 

992829 

39 

263005 

1187 

736995 

37 

24 

256523 

1146 

992806 

39 

253717 

1185 

736283 

38 

25 

257211 

1144 

992783 

39 

284428 

1183 

735572 

35 

26 

2578D8 

1142 

932759 

39 

265138 

1181 

734862 

34 

27 

258583 

1141 

992736 

39 

265847 

1179 

734153 

33 

28 

259268 

1139 

992713 

39 

266555 

1173 

733445 

32 

29 

259951 

1137 

99-2690 

39 

267261 

1176 

732739 

31 

30 

260633 

1135 

992666 

39 

267937 

1174 

732033 

30 

31 

9.261314 

1133 

9.992043 

39 

9.268671 

1172 

10.731329 

29 

32 

261994 

1131 

992619 

39 

269375 

1170 

730625 

28 

33 

262673 

1130 

992595 

39 

270077 

1109 

729923 

27 

31 

263351 

1128 

992572 

39 

270779 

1167 

729221 

26 

35 

2640-27 

1126 

99-2549 

39 

271479 

1165 

728521 

25 

36 

264703 

1124 

992525 

39 

272178 

1164 

727822 

24 

37 

265377 

1122 

99-2501 

39 

272876 

1162 

727124 

23 

38 

26(5051 

1120 

992478 

40 

273573 

1160 

726127 

22 

39 

2667-23 

1119 

992454 

40 

274259 

1158 

725731 

21 

40 

267395 

1117 

992430 

40 

274964 

1157 

725036 

20 

41 

9.268065 

1115 

9.992406 

40 

9.275658 

1155 

10.724342 

19 

42 

268734 

1113 

992382 

40 

276351 

1153 

723649 

18 

43 

269402 

1111 

992359 

40 

277043 

1151 

722957 

17 

44 

270089 

1110 

99-2335 

40 

277734 

1150 

722266 

16 

45 

270735 

1108 

992311 

40 

2784-24 

1148 

721576 

15 

48 

271400 

1106 

992287 

40 

279113 

1147 

720837 

14 

47 

272064 

1105 

992263 

40 

279801 

1145 

720199 

13 

48 

272726 

1103 

99-2-239 

40 

280488 

1143 

719512 

12 

49 

273388 

1101 

992214 

40 

281174 

1141 

718826 

11 

50 

274049 

1099 

992190 

40 

281858 

1140 

718142 

10 

51 

9.274708 

1098 

9.99216G 

40 

9.282542 

1138 

10.717458 

9 

52 

•275367 

1096 

99-2142 

40 

283225 

1136 

716775 

8 

53 

276024 

1094 

992117 

41 

283907 

1135 

716093 

7 

54 

276681 

1092 

692093 

41 

284588 

1133 

715412 

6 

55 

277337 

1091 

992059 

41 

285268 

1131 

714732 

5 

56 

277991 

108:) 

99-2044 

41 

2859  i7 

1130 

714053 

4 

57 

278644 

1087 

992020 

41 

286624 

1128 

713376 

3 

58 

279297 

1086 

99-199!) 

41 

287301 

1126 

712699 

2 

59 

279948 

1084 

991971 

41 

287977 

1125 

712023 

1 

60 

280599 

1082 

991947 

41 

288652 

1123 

711348 

0 

|   Cosine 

|    Sine   | 

Cotang.  |      |   Tang.   M. 

SINES    AND    TANGi'.NTS.       (11    DegrCCS.) 


29 


M.    Sine     D.    Cosine   |  D.  |  Tang.     D.     Cotang. 

0 

9.280599 

1082 

9.991947 

41 

9.288652 

1123 

10.711348 

60 

1 

281248 

1081 

991922 

41 

289326 

1122 

710674 

59 

2 

281897 

1079 

991897 

41 

289999 

1120 

710001 

58 

3 

282544 

1077 

991873 

4] 

290671 

1118 

709329 

57 

4 

283190 

1076 

9918-18 

41 

291342 

1117 

708658 

56 

5 

283836 

1074 

991823 

41 

292013 

1115 

707987 

55 

0 

284480 

1072 

99T99 

41 

293682 

1114 

707318 

54 

7 

285124 

1071 

991774 

42 

293350 

1112 

706650 

53 

8 

285766 

1069 

991749 

42 

294017 

1111 

705983 

52 

9 

286408 

1067 

991724 

42 

294684 

1109 

705316 

51 

10 

287048 

1066 

991699 

42 

295349 

1107 

704651 

50 

11 

9.287687 

1064 

9.991674 

42 

9.296013 

1106 

10.703987 

49 

12 

288326 

10(53 

991649 

42 

296677 

1104 

703323 

48 

13 

288964 

1061 

991624 

42 

297339 

1103 

702661 

47 

14 

289600 

1059 

991599 

42 

298001 

1101 

701999 

46 

15 

290236 

1058 

991574 

42 

298662 

1100 

701338 

45 

16 

290870 

1056 

991549 

42 

29932-3 

1098 

700678 

44 

17 

291504 

1054 

991524 

42 

299980 

1096 

700020 

43 

18 

292137 

1053 

991498 

42 

300638 

1095 

699362 

42 

11) 

292768 

1051 

991473 

42 

301295 

1093 

698705 

41 

20 

293399 

1050 

991448 

42 

301951 

1092 

698049 

40 

21 

9.294029 

1048 

9.991422 

42 

9.302607 

1090 

10.697393 

39 

22 

294658 

1046 

991397 

42 

303261 

1089 

696739 

38 

23 

295286 

1045 

991372 

43 

303914 

1087 

698086 

37 

24 

295913 

1043 

991346 

43 

304567 

1086 

695433 

36 

25 

296539 

1042 

991321 

43 

305218 

1084 

694782 

35 

26 

297164 

1040 

991295 

43 

305869 

1083 

694131 

34 

27 

297788 

1039 

991270 

43 

30(5519 

1081 

693481 

33 

28 

298412 

1037 

991244 

43 

307168 

1080 

692832 

32 

29 

299034 

1036 

991218 

43 

307815 

1078 

692185 

31 

30 

299655 

1034 

991193 

43 

308463 

1077 

691537 

30 

31 

9.300276 

1032 

9.991167 

43 

9.309109 

1075 

10.690391 

29 

32 

300895 

1031 

991141 

43 

309754 

1074 

690246 

28 

33 

301514 

10-29 

991115 

43 

310398 

1073 

689602 

27 

34 

302132 

1028 

991090 

43 

311042 

1071 

688958 

26 

35 

302748 

1026 

991064 

43 

311685 

1070 

688315 

25 

3(5 

303364 

1025 

991038 

43 

312327 

1068 

687673 

24 

37 

:«)397» 

1023 

991012 

43 

312967 

1067 

687033 

23 

38 

304593 

1022 

99098(5 

43 

313608 

1065 

686392 

22 

39 

305207 

1020 

990960 

43 

314247 

1064 

685753 

21 

40 

305819 

1019 

990934 

44 

314885 

1062 

685115 

20 

41 

9.306430 

1017 

9.990908 

44 

9.315523 

1061 

10.684477 

19 

42 

307041 

1016 

990882 

44 

316159 

1060 

683841 

18 

43 

307650 

1014 

990655 

44 

316795 

1058 

683205 

17 

44 

308259 

1013 

990829 

44 

317430 

1057 

682570 

16 

45 

308867 

1011 

990803 

44 

318064 

1055 

681936 

15 

46 

309474 

1010 

990777 

44 

318697 

1054 

681303 

14 

47 

310080 

1008 

990750 

44 

319329 

1053 

680671 

13 

48 

310685 

1007 

990724 

44 

319961 

1051 

680039 

12 

49 

311289 

1005 

iMK>97 

44 

320592 

1050 

679408 

11 

50 

311893 

1004 

990671 

44 

321222 

1048 

678778 

10 

51 

9.312495 

1003 

9.990644 

44 

9.321851 

1047 

10.678149 

9 

52 

313097 

1001 

990618 

44 

322479 

1045 

677521 

8 

53 

313698 

1000 

i«HK>9I 

44 

323106 

1044 

676894 

7 

54 

314297 

998 

990565 

44 

323733 

1043 

676267 

6 

55 

314897 

997 

990538 

44 

324358 

1041 

675642 

5 

56 

315495 

996 

99051] 

45 

324983 

1040 

675017 

4 

57 

316092 

994 

990485 

45 

325607 

1039 

674393 

3 

58 

316689 

993 

990458 

45 

326231 

1037 

673769 

2 

59 

317284 

991 

990431 

45 

326853 

1036 

673141 

1 

GO 

317879 

990 

<i!H!ini 

45 

327475 

1035 

672525 

0 

Cosine          Sine   |    I  Cotang.  |         Tang.    M. 

78  Degrees. 


30 


(12  Degrees.)     A  TABLE  or  LOGARITHMIC 


i  M    Sine      D     Cosine  |  D.  |  Tang.   |   D.   |  Cotang.  | 

0 

9.317879 

990 

9.990404 

45 

9.327474 

1035 

10.672526 

60 

1  1 

318473 

988 

99U378 

45 

328095 

1033 

671905 

.V.I 

i  2 

319066 

987 

990351 

45 

328715 

1032 

671285 

58 

3 

319658 

986 

990324 

45 

329334 

1030 

670666 

57 

4 

320249 

984 

990297 

45 

329953 

1G29 

670047 

56 

5 

320840 

983 

990-270 

45 

330570 

1028 

GT9430 

55 

G 

321430 

982 

990243 

45 

331187 

1026 

668813 

54 

7 

322019 

980 

990215 

45 

:mso3 

1025 

668197 

53 

8 

322C07 

979 

990188 

45 

332418 

1024 

667582 

52 

9 

323194 

977 

990161 

45 

333033 

1023 

G6f967 

51 

10 

323780 

976 

990134 

45 

333646 

1021 

666354 

50 

11 

9.324366 

975 

9.990107 

46 

9.334259 

1020 

10.665741 

49 

12 

324950 

973 

990079 

46 

334871 

10J9 

665129 

48 

13 

325534 

972 

99QC52 

46 

335482 

1017 

664518 

47 

14 

326117 

970 

990025 

46 

336093 

1016 

663907 

46 

15 

326700 

969 

989997 

46 

336702 

1015 

663298 

45 

1  16 

327281 

968 

989970 

46 

337311 

1013 

662689 

44 

1  I? 

327862 

966 

989942 

46 

337919 

1012 

662081 

43 

18 

328442 

965 

989915 

46 

338527 

1011 

661473 

42 

19 

329021 

964 

989887 

46 

339133 

1010 

660867 

41 

20 

329599 

962 

989860 

46 

339739 

1008 

600261 

40 

21 

9.330176 

961 

9.989832 

46 

9.340344 

1007 

10.659056 

39 

22 

330753 

960 

989804 

46 

340948 

1006 

(559052 

38 

23 

331329 

958 

989777 

46 

341552 

1004 

658448 

37 

24 

331903 

957 

989749 

47 

342155 

1003 

65?  845 

36 

25 

332478 

956 

989721 

47 

342757 

1002 

657243 

35 

26 

333051 

954 

989693 

47 

343358 

1000 

050042 

34 

27 

333624 

953 

9896C5 

47 

342958 

999 

656042 

33 

28 

334195 

952 

989637 

47 

344558 

998 

655442 

32 

29 

334766 

950 

989609 

47 

345157 

997 

654843 

31 

30 

335337 

949 

989582 

47 

345755 

996 

654245 

30 

31 

9.335906 

948 

9.989553 

47 

9.340353 

994 

10.C53647 

29 

32 

336475 

946 

989525 

47 

34(5949 

993 

653051 

28 

33 

337043 

945 

9894SI7 

47 

347545 

992 

652455 

27 

34 

337610 

944 

989469 

47 

348141 

991 

651859 

26 

35 

338176 

943 

989441 

47 

348735 

990 

651265 

25 

36 

338742 

941 

989413 

47 

349329 

988 

650671 

24 

37 

339306 

940 

989384 

47 

349922 

987 

650078 

23 

38 

339871 

939 

989356 

47 

350514 

986 

649486 

22 

39 

340434 

937 

989328 

47 

351106 

985 

648894 

21 

40 

340996 

936 

989300 

47 

351697 

983 

648303 

20 

41 

9.341558 

935 

9.989271 

47 

9.352287 

982 

10.647713 

19 

42 

342119 

934 

989243 

47 

352876 

981 

647124 

18 

43 

342679 

932 

989214 

47 

353465 

980 

646535 

17 

44 

343239 

931 

989186 

47 

354053 

979 

645947 

16 

45 

343797 

930 

989157 

47 

354640 

977 

645360 

15 

46 

344355 

929 

989128 

48 

355227 

976 

644773 

14 

47 

344912 

927 

989100 

48 

355813 

975 

C44187 

13 

48 

345469 

926 

989071 

48 

356398 

974 

643602 

12 

49 

346024 

925 

989042 

48 

356982 

973 

643018 

11 

50 

346579 

924 

989014 

48 

357566 

971 

642434 

10 

51 

9.347134 

922 

9.988985 

48 

9.358149 

970 

10.641851 

9 

52 

347687 

921 

988956 

48 

358731 

969 

641269 

8 

53 

348240 

Q-20 

988927 

48 

359313 

968 

640C87 

7 

54 

348792 

919 

988898 

48 

359893 

967 

640107 

6 

53 

349343 

917 

988869 

48 

360474 

966 

639526 

5 

56 

349893 

916 

988840 

48 

361053 

965 

638947 

4 

57 

350443 

915 

988811 

49 

361632 

963 

638308 

3 

58 

350992 

914 

988782 

49 

362210 

962 

637790 

2 

59 

351540 

913 

988753 

49 

332787 

961 

637213 

1 

60 

352088 

911 

988724 

49 

363364 

960 

636636 

0 

Cosine 

|   Sine  |    |  Cotang.  |      |   Tang. 

M. 

77  Degrees. 


SINES  AND  TANGENTS.     (13  Degrees.) 


M.    Sine  |   D.  |  Cosine  |  D.   Tang.     D.    Coking. 

0 

!».:C,JIIH,S 

911 

9.988724 

49 

J».:n53:i'i  i 

960 

10.636(536 

60 

1 

352835 

910 

988895 

49 

383940 

959 

036060 

59 

2 

353181 

909 

988666 

49 

364515 

9.18 

635485 

58 

9 

:i:>:!?-j(> 

908 

'   988(536 

49 

365090 

957 

634910 

57 

4 

35427J 

go? 

988807 

49 

3(55664 

955 

634336 

56 

5 

354815 

905 

988578 

49 

366237 

9->4 

6337(53 

55 

0 

355358 

904 

988548 

49 

366810 

953 

633190 

54 

7 

355901 

903 

988519 

49 

367382 

999 

632(518 

53 

8 

356443 

902 

938489 

49 

367953 

951 

632047 

52 

n 

901 

988460 

49 

368524 

950 

631476 

51 

10 

357521 

899 

988430 

49 

3li!H)'Jl 

'.(4!) 

630906 

50 

u 

9.3580r>4 

898 

9.  98*401 

43 

9.3696i53 

948 

10.630337 

49 

12 

358803 

897 

988371 

49 

370232 

946 

621)708 

48 

13 

359141 

896 

988342 

49 

370799 

945 

629201 

47 

14 

359678 

895 

988312 

50 

371367 

944 

628633 

46 

15 

360215 

893 

988282 

50 

371933 

943 

628067 

45 

16 

360752 

892 

988252 

50 

372499 

942 

627501 

44 

17 

361287 

891 

988223 

50 

373064 

941 

626936 

43 

18 

361822 

890 

988193 

50 

373629 

940 

626371 

42 

19 

3(12356 

889 

988163 

50 

374193 

939 

625807 

41 

20 

362889 

888 

988133 

50 

374756 

938 

625244 

40 

21 

9.3(53422 

887 

9.988103 

50 

9.375319 

937 

10.624681 

39 

22 

303U54 

885 

988073 

50 

375881 

935 

624119 

38 

23 

364485 

884 

988J43 

50 

37(5442 

934 

623558 

37 

24 

3G5016 

883 

988013 

50 

377003 

933 

622997 

36 

25 

365546 

882 

987983 

50 

377563 

932 

622437 

35 

26 

31)0075 

881 

987953 

50 

378122 

931 

621878 

34 

27 

3IMU504 

880 

987922 

50 

378681 

930 

621319 

33 

28 

3157131 

879 

987892 

50 

379239 

929 

620761 

32 

29 

367659 

877 

987862 

50 

379797 

928 

620203 

31 

30 

368185 

876 

987832 

51 

380354 

927 

619646 

30 

31 

9.3(58711 

875 

9.987801 

51 

9.380910 

926 

10.619090 

29 

32 

3*59236 

874 

987771 

51 

381466 

925 

618534 

28 

-S3 

3697(51 

873 

987740 

51 

382020 

924 

617980 

27 

34 

370285 

872 

987710 

51 

38-2575 

923 

617425 

26 

35 

370808 

871 

987679 

51 

383129 

922 

616871 

25 

36 

371330 

870 

987649 

51 

383(582 

921 

616318 

24 

37 

371852 

869 

987618 

51 

384234 

920 

615766 

23 

38 

372373 

867 

'  987588 

51 

384786 

919 

615214 

22 

39 

372894 

866 

987557 

51 

385337 

918 

614663 

21 

40 

373414 

865 

987526 

51 

385888 

917 

614112 

20 

41 

9.373933 

864 

9.987496 

51 

9.386438 

915 

10.613562 

19 

42 

374452 

863 

987465 

51 

38(5987 

914 

613013 

18 

43 

374970 

862 

987434 

51 

387536 

913 

612464 

17 

44 

375487 

861 

987403 

52 

388084 

912 

611916 

16 

45 

37(iO()3 

860 

987372 

52 

388631 

911 

611369 

15 

46 

376519 

859 

9^7341 

52 

389178 

910 

610822 

14 

47 

377035 

858 

987310 

52 

389724 

909 

610276 

13 

48 

377549 

857 

987279 

52 

390270 

908 

609730 

12 

49 

378063 

856 

987248 

52 

390815 

907 

609185 

11 

50 

378577 

854 

987217 

52 

391360 

906 

608640 

10 

51 

9.379089 

853 

9.987186 

52 

9.391903 

905 

10.608097 

9 

52 

379601 

852 

987155 

52 

392447 

9i)4 

(Tl  17553 

8 

53 

380113 

851 

987124 

52 

392989 

903 

607011 

7 

54 

380624 

850 

9&7092 

52 

:w:r>:u 

902 

6015469 

6 

55 

381134 

849 

9870(51 

52 

394073 

901 

605U27 

5 

5G 

38*643 

848 

987030 

52 

394614 

900 

605386 

4 

57 

382152 

847 

9855998 

52 

395154 

899 

604846 

3 

58 

3H-J661 

846 

986967 

52 

3!>."i''.94 

898 

604306 

2 

59 

383168 

845 

986936 

52 

396233 

897 

6037(57 

1 

60 

383675 

844 

98^904 

52 

396771 

896 

603229 

0 

|  Cosiiu-  |      |   Sine   |    |  Cotang.  |      |    Tang.    M. 

76  Degrees. 


(14  Degrees.)    A  TABLE  OF  LOGARITHMIC 


M.    Sine     D   j  Cosine    D.  |  Tang.   |   D. 

Cotang.  | 

° 

9.383675 

844 

9.986904 

52 

9.396771 

896 

10.603229   60 

1 

384182 

843 

986873 

53 

397309 

896 

602691   .TO 

2 

384687 

842 

986841 

53 

397846 

895 

602154 

58 

3 

385192 

841 

980809 

53 

398383 

894 

G01617 

57 

4 

385697 

840 

986778 

53 

3i)8<m) 

893 

601081 

56 

5 

38(5201 

839 

986746 

53 

399455 

892 

600545 

55 

6 

386704 

838 

986714 

53 

399990 

891 

600010 

54 

7 

387207 

837 

986683 

53 

400534 

890 

599476 

53 

8 

387709 

836 

986651 

53 

401058 

889 

598942 

52 

9 

388210 

835 

986619 

53 

401591 

888 

598409 

51 

10 

388711 

834 

986587 

53 

402124 

887 

887878 

50 

11 

9.389211 

833 

9.986555 

53 

9.402S55 

886 

10.597344 

49 

12 

389711 

832 

986C--23 

53 

403187 

885 

596813 

48 

13 

390210 

831 

986491 

53 

403718 

884 

596282 

47 

14 

390708 

830 

986459 

53 

404249 

883 

595751 

46 

15 

SSI206 

828 

986427 

53 

404778 

882 

595222 

45 

J*^ 

391703 

827 

986395 

53 

405308 

881 

594692 

44 

17 

392199 

826 

986363 

54 

405836 

880 

594164 

43 

18 

392695 

825 

986331 

54 

406364 

879 

593636 

42 

19 

393191 

824 

986299 

54 

406892 

878 

593108 

41 

20 

393G85 

823 

986266 

54 

407419 

877 

592581 

40 

21 

9.394179 

822 

9.986234 

54 

9.407945 

876 

10.592055 

39 

22 

394673 

821 

986202 

54 

408471 

875 

591529 

38 

23 

395166 

820 

986169 

54 

408997 

874 

591003 

37 

24 

395658 

819 

986137 

54 

409521 

874 

590479 

36 

25 

396150 

818 

986104 

54 

410045 

873 

589955 

35 

26 

396641 

817 

986072 

54 

4105139 

872 

589431 

54 

27 

397132 

817 

986039 

54 

411092 

871 

588908 

33 

28 

397621 

816 

986007 

54 

411615 

870 

588385 

32 

29 

398111 

815 

985974 

54 

412]  37 

869 

587863 

31 

30 

398600 

814 

985942 

54 

412658 

808 

587342 

30 

31 

9.399088 

813 

9.985909 

55 

9.413179 

867 

10.586821 

29 

32 

399575 

812 

985876 

55 

413699 

866 

586301 

28 

33 

400062 

811 

985843 

55 

414219 

865 

585781 

27 

34 

400549  i  810 

985811 

55 

414738 

864 

585262 

2<f 

35 

401035 

809 

985778 

55 

415257 

864 

584743 

25 

36 

401520 

808 

985745 

55 

415775 

863 

584225 

24 

37 

402005 

807 

985712 

55 

416293 

862 

583707 

23 

38 

402-189 

806 

985679 

55 

416810 

861 

583190 

22 

39 

402972 

805 

985646 

55 

417326 

860 

582674 

21 

40 

403455 

804 

985613 

55 

417842 

859 

582158 

20 

41 

9.403938 

803 

9.985580 

55 

9.418358 

858 

10.581642 

19 

42 

404420 

802 

985547 

55 

418873 

857 

581127 

18 

43 

404901 

801 

985514 

55 

419387 

856 

580613 

17 

44 

405382 

800 

985480 

55 

419901 

855 

580099 

16 

45 

405862 

799 

965447 

55 

420415 

855 

579585 

15 

46 

406341 

798 

985414 

56 

420927 

854 

579073 

14 

47 

406820 

797 

985380 

56 

421440 

853 

578560 

13 

48 

407299 

796 

985347 

56 

421952 

852 

578048 

12 

49 

407777 

795 

985314 

56 

422463 

851 

577537 

11 

50 

408254 

794 

985280 

56 

422974 

850 

577026 

10 

51 

9.408731 

794 

9.985247 

56 

9.423484 

849 

10.576516 

9 

52 

409207 

793 

985213 

56 

423993 

848 

576007 

R 

53 

409682 

792 

985180 

56 

424503 

848 

575497 

7 

54 

410157 

791 

985146 

56 

425011 

847 

574989 

6 

55 

410632 

790 

985113 

56 

425519 

846 

574481 

5 

56 

411106 

789 

985079 

56 

426027 

845 

573973 

4 

57 

411579 

788 

985045 

56 

426534 

844 

573466 

3 

58 

412052 

787 

985011 

56 

427041 

843 

572959 

2 

59 

412524 

786 

984978 

56 

427547 

843 

572453 

1 

60 

412996 

785 

984944 

56 

428052 

842 

571948 

0 

Cosine       |   Sine   |    |   Cotang.  |         Taiig.    M. 

75  Degrees. 


SINES  AND  TANGENTS.     (15  Degrees.) 


33 


M.  |   Sine   |   D.  |  Cosine  |  D.  |   Tang.   |   D.  |  Cotang.  |   1 

0 

<j.4i--".nm 

785 

57 

9.438052 

842 

10.571948 

60  J 

1 

4134157 

784 

984910 

57 

428557 

841 

571443 

59 

2 

413938 

783 

934B70 

57 

42i)062 

840 

570938 

58 

3 

414408 

7,^3 

984842 

57 

429566 

839 

570434 

57 

4 

414878 

782 

984808 

57 

430070 

838 

569930 

50 

5 

415347 

781 

984774 

57 

430573 

838 

509427 

55 

6 

41W15 

780 

984740 

57 

431075 

837 

508925 

51 

7 

416383 

77'.) 

984706 

57 

431577 

836 

508423 

53 

8 

41675] 

778 

984673 

57 

433079 

835 

567921 

52 

9 

417217 

777 

984637 

57 

432580 

834 

567420 

51 

10 

417684 

776 

984603 

57 

433080 

833 

566920 

50 

11 

9.418150 

775 

9-984569 

r>7 

9.433580 

832 

10.50(5420 

49 

12 

418G15 

774 

984535 

57 

434080 

832 

565920 

48 

13 

419079 

773 

984500 

57 

434579 

831 

565421 

47 

14 

419544 

773 

984406 

57 

435078 

830 

564922 

46 

15 

4-201)07 

772 

984433 

58 

435576 

829 

5(54424 

45 

16 

420470 

771 

984397 

58 

•)  36073 

828 

563927 

44 

17 

420933 

770 

984363 

56 

436570 

828 

563430 

43 

18 

421395 

769 

984328 

58 

437067 

827 

562933 

42 

19 

421857 

768 

984294 

58 

437563 

826 

562437 

41 

2U 

422318 

767 

984259 

58 

438059 

825 

561941 

40 

21 

9.422778 

767 

9.984224 

58 

9.438554 

824 

10.561446 

39 

2-2 

423238 

768 

984190 

58 

439048 

823 

560952 

38 

23 

423097 

765 

984155 

58 

439543 

823 

560457 

37 

24 

424156 

764 

984120 

58 

440036 

822 

559964 

36 

25 

424615 

703 

984085 

58 

440529 

821 

559471 

35 

20 

425073 

762 

984050 

58 

411022 

820 

558978 

34 

27 

425530 

701 

9840J5 

53 

441514 

819 

558486 

33 

•J- 

425987 

7CO 

58 

442006 

819 

557994 

:w 

29 

4*26443 

780 

983946 

58 

442497 

818 

557503 

31 

30 

426899 

759 

983911 

58 

442988 

817 

557012 

30 

31 

9.427354 

758 

9.983875 

58 

9.443479 

816 

10.550521 

29 

:w 

427809 

757 

983840 

59 

443968 

816 

556032 

28 

33 

428963 

756 

983805 

59 

444458 

815 

555542 

27 

34 

428717 

755 

983770 

59 

444947 

814 

555053 

26 

35 

429170 

754 

983735 

59 

445435 

813 

554565 

25 

30 

429623 

753 

983700 

59 

445923 

812 

554077 

24 

37 

430075 

752 

983664 

59 

446411 

812 

553589 

23 

38 

430527 

753 

983629 

59 

446898 

811 

553102 

22 

39 

430978 

7.51 

983594 

59 

447384 

810 

552616 

21 

40 

431429 

750 

983558 

59 

447870 

809 

552130 

20 

41 

9.431879 

749 

9.983523 

59 

9.448356 

809 

10.551(544 

19 

42 

432329 

749 

983487 

59 

448841 

808 

551159 

18 

43 

432778 

748 

983452 

59 

449326 

807 

550674 

17 

44 

433226 

747 

983416 

59 

449810 

806 

'  550190 

16 

45 

433675 

746 

983381 

59 

450294 

806 

549706 

15 

46 

434122 

745 

983345 

59 

450777 

805 

549223 

14 

47 

434569 

744 

983309 

59 

451260 

804 

548740 

13 

48 

435016 

744 

9&3273 

60 

451743 

803 

548257 

12 

49 

435462 

743 

983238 

60 

452225 

802 

547775 

11 

50 

435908 

742 

983202 

60 

452706 

802 

547294 

10 

51 

9-436353 

741 

9.983166 

60 

9.453187 

801 

10.546813 

9 

52 

436798 

740 

983130 

60 

453668 

800 

546332 

8 

53 

437242 

740 

983094 

60 

454148 

799 

545852 

7 

54 

437686 

739 

983058 

60 

454628 

799 

5-15372 

6 

55 

438129 

738 

983022 

60 

455107 

798 

544893 

5 

56 

438572 

737 

982988 

60 

455586 

797 

544414 

4 

57 

439014 

730 

982950 

60 

456064 

796 

543936 

3 

58 

439456 

738 

982914 

60 

456542 

796 

543458 

2 

59 

439897 

735 

982878 

60 

457019 

795 

542981 

i  ! 

00 

440338 

734 

982842 

60 

457496 

794 

5-12504 

0 

|  Cosine 

Sine   I    |  Cotang.  |      |   Tang.  |  M. 

74  Degrees. 

2* 

34 


(16  Degrees.)     A  TABLE  or  LOGARITHMIC 


M  !   Sine     D.   |  Cosine  |  D.  |   Tang.  |   D     Cotang. 

0 

9.440338 

734 

9.9*-'.-4-2 

1)0 

9.4.r>T4:)li 

794 

10.54-2.104 

60 

1 

440778 

733 

982805 

60 

457973 

793 

542027 

59 

2 

441218 

732 

982769 

61 

458449 

793 

541551 

58 

3 

441658 

731 

982733 

61 

458925 

792 

541075 

57 

4 

442096 

731 

98-2690 

61 

4594:)0 

791 

540600 

56 

5 

442535 

730 

98-2600 

61 

459875 

790 

540125 

55 

6 

442973 

789 

982624 

61 

400349 

790 

539051 

54 

7 

44:5410 

728 

982587 

61 

460823 

789 

539177 

53 

8 

443847 

727 

982551 

61 

461297 

788 

538703 

52 

9 

444284 

727 

<J.~2.-)14 

61 

461770 

788 

538230 

51 

10 

444720 

726 

982477 

61 

462242 

787 

537758 

50 

11 

9.445155 

725 

9.98-2441 

61 

9.402714 

786 

10.537286 

49 

12 

445590 

724 

982404 

61 

403186 

785 

53(5814 

48 

13 

446025 

723 

982367 

61 

463658 

785 

530342 

47 

14 

446459 

723 

982331 

61 

464129 

784 

535871 

46 

15 

446893 

722 

982294 

61 

464599 

783 

535401 

45 

16 

4473-26 

721 

982-257 

61 

465069 

783 

534931 

44 

:  17 

447759 

720 

982220 

62 

465539 

782 

534461 

43 

18 

448191 

720 

982183 

62 

466008 

781 

533992 

42 

19 

448623 

719 

982146 

62 

4G6476 

780 

533524 

41 

20 

449054 

7J8 

98-2109 

62 

466945 

780 

533055 

40 

21 

9.449485 

717 

9.982072 

62 

9.467413 

779 

10.532587 

39 

"ft 

449915 

716 

982035 

62 

467880 

778 

532120 

38 

23 

450345 

716 

981998 

62 

468347 

778 

531653 

37 

24 

450775 

715 

981961 

62 

468814 

777 

531186 

36 

25 

451204 

714 

981924 

62 

469280 

776 

530720 

35 

20 

451632 

713 

981885 

62 

40974G 

775 

530254 

34 

27 

452060 

713 

981849 

62 

4/u211 

775 

529789 

33 

28 

452488 

712 

981812 

62 

470G76 

V74 

529324 

32 

29 

452915 

711 

981774 

62 

471141 

773 

6-23859   31 

30 

453342 

710 

981737 

62 

471605 

773 

528395 

iwi  , 

31 

9.453768 

710 

9.  98  1099 

63 

9.472068 

772 

10.527932 

29 

32 

454194 

709 

-  981662 

63 

47-25.32 

771 

527468 

28 

33 

454619 

708 

981625 

63 

472995 

771 

52.005 

27 

34 

455044 

707 

981587 

63 

473457 

770 

526543 

26 

35 

455469 

707 

981549 

63 

473919 

769 

524)081 

25 

36 

455893 

706 

981512 

63 

474381 

769 

525619 

24 

37 

4563]  6 

705 

981474 

63 

474842 

768 

525158 

23 

38 

456739 

704 

981436 

63 

475303 

767 

524697 

22 

39 

457162 

704 

981399 

63 

475763 

707 

524237 

21 

40 

457584 

703 

981361 

63 

476223 

766 

523777 

20 

41 

9.458006 

702 

9.981323 

63 

9.476683 

765 

10.523317 

19 

42 

458427 

701 

981285 

63 

477142 

765 

522858 

18 

43 

458848 

701 

981247 

63 

477601 

764 

522399 

17 

44 

4592<58 

700 

981209 

63 

478059 

763 

521941 

16 

1  4."> 

459688 

699 

981171 

63 

478517 

763 

521483 

15 

46 

460108 

698 

981133 

64 

478975 

762 

521025 

14 

47 

460527 

698 

981095 

64 

479432 

761 

520568 

13 

48 

46094«K 

697 

981057 

64 

479889 

701 

520111 

12 

49 

461364 

698 

981019 

64 

480345 

760 

519655 

11 

50 

461782 

695 

980981 

64 

480801 

759 

519199 

10 

51 

9.462199 

695 

9.980942 

64 

9.481257 

759 

10.513743 

9 

52 

462616 

694 

980904 

64 

481712 

758 

518288 

8 

53 

463032 

693 

980866 

64 

482167 

757 

517833 

7 

54 

463448 

693 

980827 

64 

482621 

7f>7 

517379 

6 

55 

463864 

692 

980789 

64 

483075 

756 

516925 

5 

56 

464279 

691 

980750 

64 

483529 

755 

510471 

4 

57 

464694 

690 

980712 

64 

483982 

755 

516018 

3 

58 

465108 

690 

980673 

64 

484435 

754 

515565 

2 

59 

465522 

689 

980635 

64 

484887 

753 

515113 

1 

60 

465935 

688 

980596 

64 

485339 

753 

514661 

0 

|  Cosine  |      |   Sine   |    |  Cotang.       |   Tang.   |  M.  1 

73  Degrees. 


SINES    AND    TANGENTS.       (17  Degrees.) 


35 


i  M.  |   Sine     D.   |   Cosine   D.    Tang.     D.  |  Cotanj?.      ' 

0 

9.4I>5:I35 

BBS 

9.980596 

64 

9.485339 

V>5 

10.5l4(i()l 

60 

1 

460318 

688 

080558 

64 

485791 

752 

514209 

59 

o 

466781 

687 

930519 

65 

480242 

751 

513758 

58 

3 

•167173 

686 

980480 

65 

4Si)fi!)3 

751 

513307 

57 

4 

467585 

685 

9804452 

65 

487143 

750 

512857 

56 

5 

487996 

685 

060403 

65 

487593 

749 

512407 

55 

6 

468407 

684 

9803(54 

65 

488043 

749 

511957 

54 

1 

468817 

883 

080325 

65 

488492 

748 

511508 

53 

B 

469-227 

683 

989286 

65 

488941 

747" 

511059 

52 

9 

469U37 

682 

980947 

65 

489390 

747 

510510 

51 

10 

470046 

681 

980208 

65 

489838 

746 

510162 

50 

11 

9.470455 

680 

9.980169 

65 

9.490236 

746 

10.509714 

49 

12 

470863 

689 

980130 

63 

490733 

745 

509267 

48 

13 

471271 

679 

980091 

65 

491180 

744 

508820 

47 

14 

471679 

678 

980052 

65 

491627 

744 

508373 

46 

15 

472086 

678 

930012 

65 

492073 

743 

507927 

45 

16 

472492 

677 

979973 

65 

492519 

743 

507481 

44 

17 

47-2898 

676 

979934 

66 

492965 

742 

507035 

43 

18 

473304 

676 

979895 

66 

493410 

741 

508590 

42 

19 

473710 

675 

979855 

66 

493854 

740 

506146 

41 

20 

474115 

674 

979816 

66 

494299 

740 

505701 

40 

21 

9-474519 

674 

9.979776 

66 

9.494743 

740 

10.505257 

39 

33 

474923 

673 

979737 

66 

495186 

739 

504814 

38 

23 

475327 

672 

979G97 

66 

495630 

738 

504370 

37 

24 

475730 

672 

979558 

60 

496073 

737 

503927 

36 

25 

476133 

671 

979618 

m 

496515 

737 

503485 

35 

26 

476536 

670 

979579 

66 

496957 

736 

503043 

34 

27 

476938 

669 

979539 

66 

497399 

736 

502G01 

33 

28 

477340 

669 

979499 

63 

497841 

735 

502159 

32 

29 

477741 

668 

979459 

66 

498282 

T34 

501718 

31 

30 

478142 

667 

979420 

t>6 

498722 

734 

501278 

30 

31 

9.478542 

667 

9.979380 

66 

9.499163 

733 

10.500837 

29 

3-2 

478942 

666 

979340 

66 

499603 

733 

500397 

28 

33 

479342 

665 

979300 

67 

500042 

732 

499958 

27 

34 

479741 

665 

9792(50 

67 

500481 

731 

499519 

26 

35 

480140 

664 

979220 

67 

500920 

731 

499080 

25 

36 

480539 

663 

979180 

67 

501359 

730 

498;i41 

24 

37 

480437 

663 

979140 

67 

501797 

730 

498203 

23 

38 

481334 

662 

979100 

67 

502-235 

729 

497765 

22 

39 

481731 

661 

979059 

67 

50-2672 

728 

497328 

21 

40 

482128 

661 

979019 

67 

503109 

728 

493891 

20 

41 

9.482525 

660 

9.978979 

67 

9.503546 

727 

10.496454 

19 

42 

482921 

659 

978939 

67 

503982 

727 

496018 

18 

43 

483316 

659 

978898 

67 

504418 

726 

495582 

17 

44 

483712 

658 

978858 

67 

504854 

725 

495146 

16 

45 

484107 

657 

978817 

67 

505289 

725 

494711 

15 

46 

484501 

657 

978777 

67 

505724 

724 

494276 

14 

47 

484895 

656 

978736 

67 

506159 

724 

493841 

13 

48 

485289 

655 

978696 

68 

50(5593 

723 

493407 

12 

49 

485682 

655 

978655 

63 

507027 

722 

492973 

11 

50 

486075 

654 

978615 

68 

507460 

722 

492540 

10 

51 

9.486467 

653 

9.978574 

68 

9.507893 

721 

10.492107 

9 

5-1 

488860 

653 

978533 

68 

508326 

721 

491674 

8 

53 

487251 

652 

978493 

68 

508759 

720 

491241 

7 

54 

487643 

651 

978452 

68 

509191 

719 

490809 

6 

55 

488034 

651 

978411 

68 

509622 

719 

490378 

5 

56 

4HH4-24 

650 

978370 

68 

510054 

718 

489946 

4 

57 

4888J4 

650 

978329 

68 

510485 

718 

489515 

3 

58 

489204 

649 

978288 

68 

510916 

717 

489084 

2 

59 

483593 

648 

978247 

68 

511346 

716 

488654 

1 

60 

489932 

648 

978206 

68 

51L776 

716 

4H8-224 

0 

|  Cosine  |      |   Sine   |    |  Cotang.  |      |   Tang.    M. 

72  Degrees. 


(18  Degrees.)     A  TABLE  or  LOGARITHMIC 


M. 

Sine 

D 

Cosine 

D. 

!  Tang. 

!  D. 

|  Cotang. 

0 

9  489982 

648 

9.978206 

68 

9.511776 

716 

10.488224 

60 

1 

490371 

648 

978165 

68 

512206 

718 

487794 

59 

2 

490759 

647 

978124 

68 

512(535 

7J5 

487365 

58 

3 

491147 

046 

978083 

69 

513064 

714 

48G936 

57 

4 

491535 

646 

978042 

69 

513493 

714 

486507 

56 

5 

491922 

645 

978001 

69 

513921 

713 

486079 

55 

6 

49230? 

644 

977959 

69 

514349 

713 

485651 

54 

7 

49269J 

644 

977918 

69 

514777 

712 

485223 

53 

8 

493081 

H43 

977877 

69 

515204 

712 

484796 

52 

9 

493466 

642 

977835 

69 

515631 

711 

484369 

51 

10 

493851 

642 

977794 

69 

516057 

710 

483943 

50 

11 

9.494236 

.641 

9.977752 

69 

9.516484 

710 

10.483516 

49 

12 

494621 

641 

977711 

69 

516910 

709 

483090 

48 

13 

495005 

640 

9776G9 

69 

517335 

709 

482665 

47 

14 

495388 

639 

977628 

69 

517761 

708 

482239 

46 

15 

495772 

639 

977586 

69 

518185 

708 

481815 

45 

16 

496154 

638 

977544 

70 

518610 

707 

481390 

44 

17 

496537 

637 

977503 

70 

519034 

706 

480966 

43 

18 

496919 

637 

977461 

70 

519458 

70G 

480542 

42 

19 

497301 

636 

977419 

70 

519882 

705 

480118 

41 

20 

497682 

636 

977377 

70 

520305 

705 

479CD5 

40 

21 

9.498064 

635 

9.977335 

70 

9.520728 

704 

10.479272 

ao 

22 

498444 

634 

977293 

70 

521151 

703 

478849 

38 

23 

498825 

634 

977251 

70 

521573 

703 

478427 

37 

24 

499204 

633 

977209 

70 

521995 

703 

478005 

36 

25 

499584 

632 

977167 

70 

522417 

7G2 

77583 

33 

26 

499963 

632 

9771-25 

70 

52-3838 

702 

77JG2 

::: 

27 

500342 

631 

977083 

70 

523259 

701 

76741 

33 

28 

500721 

631 

977041 

70 

523(!8» 

701 

70320 

32 

29 

501099 

630 

U76999 

70 

524100 

700 

75900 

31 

30 

501476 

629 

97G957 

70 

524520 

699 

75480 

30 

31 

9.501854 

629 

9.976914 

70 

9.524339 

699 

10.475061 

29 

32 

502231 

628 

976872 

71 

525359 

698 

474641 

28 

33 

502607 

628 

976830 

71 

525778 

698 

474222 

27 

34 

502984 

627 

970787 

71 

5-2I.J97 

697 

473803 

26 

35 

50;i3GO 

626 

976745 

71 

526G15 

697 

473385 

25 

36 

503735 

626 

976702 

71 

527033 

696 

472967 

24 

37 

504110 

625 

970000 

71 

527451 

696 

472549 

23 

38 

504485 

625 

976617 

71 

527868 

695 

472132 

22 

39 

504860 

624 

976574 

71 

528285 

695 

471715 

21 

40 

505234 

623 

976532 

71 

528702 

694 

471298 

20 

41 

9.505608 

623 

9.976489 

71 

9.529119 

693 

10.470831 

19 

42 

505981 

622 

976446 

71 

529535 

693 

470405 

18 

43 

506354 

622 

976404 

71 

529950 

693 

470050 

17 

44 

506727 

621 

976361 

71 

530366 

692 

409634 

16 

45 

507099 

620 

976318 

71 

530781 

691 

4(59219 

15 

46 

507471 

620 

976275 

71 

531196 

691 

468804 

14 

47 

507843 

619 

976232 

72 

531611 

690 

468389 

13 

48 

508214 

619 

976189 

72 

532021 

690 

467975 

12 

49 

508585 

618 

976146 

72 

532439 

689 

467561 

11 

50 

508956 

618 

976103 

72 

532853 

689 

467147 

10 

51 

9.509326 

617 

9-976060 

72 

9.533266 

688 

10.466734 

9 

52 

509696 

616 

976017 

72 

533679 

688 

466321 

8 

53 

510065 

616 

975974 

72 

534092 

687 

465908 

7 

54 

510434 

615 

975930 

72 

534504 

687 

465496 

6 

55 

510803 

615 

975887 

72 

534916 

686 

465084 

5 

56 

511172 

614 

975844 

72 

535328 

686 

464672 

4 

57 

511540 

613 

975800 

72 

535739 

685 

464261 

3 

58 

511907 

613 

975757 

72 

536150 

685 

463850 

2 

59 

512275 

612 

975714 

72 

536561 

684 

463439 

1 

60 

512642 

612 

975670 

72 

536972 

684 

463028 

0 

Cosine  | 

I 

Sine  | 

Cotang.  | 

1 

Tang.   | 

M. 

71  Degrees. 


SINES  AND  TANGENTS.     (19  Degrees.) 


37 


M.  |   Sine   |   D.   |  Cosine  |  D.  |  Tang.     D. 

Cotang.  | 

0 

9.512IU2 

612 

9.97.5670 

73 

9.530972   084 

10.  463028 

eo 

1 

513009 

611 

975627 

73 

537383 

683 

462018 

59 

a 

:>  1:075 

611 

975583 

73 

537792 

683 

462208 

58 

3 

513741 

610 

975539 

73 

538202 

682 

461798 

57 

4 

514107 

609 

975496 

73 

53861] 

682 

401389 

56 

5 

514472 

609 

975452 

73 

539020 

681 

460980 

55 

6 

5  14SU7 

608 

975408 

73 

539429 

681 

460571 

54 

7 

515202 

608 

975365 

73 

539837" 

680 

460163 

53 

8 

5155(56 

607 

975321 

73 

540245 

680 

459755 

52 

9 

515930 

607 

975277 

73 

540653 

679 

459347 

51 

10 

510294 

60(5 

975233 

73 

5-1  1  (Mil 

679 

458939 

50 

11 

9.516657 

605 

9.975189 

73 

9.541468 

678 

10.458532 

49 

12 

517020 

605 

975145 

73 

541875 

678 

458125 

48 

13 

517382 

604 

975101 

73 

542281 

677 

457719 

47 

14 

517745 

604 

975057 

73 

542688 

677 

457312 

46 

15 

518107 

603 

975013 

73 

543094 

676 

456906 

45 

16 

518468 

603 

974909 

74 

543499 

676 

450501 

44 

17 

518829 

602 

974925 

74 

543905 

675 

456095 

43 

18 

519190 

601 

974880 

74 

544310 

675 

455690 

42 

19 

519551 

601 

974836 

74 

544715 

674 

455285 

41 

20 

519911 

600 

974792 

74 

545119 

674 

454881 

40 

21 

9.520271 

600 

9.974748 

74 

9.545524 

673 

10.454476 

39 

22 

520031 

599 

974703 

74 

545928 

673 

454072 

38 

23 

520990 

599 

974659 

74 

546331 

672 

453(569 

37 

24 

521349 

598 

974014 

74 

546735 

672 

453265 

36 

25 

521707 

598 

974570 

74 

547138 

671 

452862 

35 

98 

539066 

597 

974525 

74 

547540 

671 

452460 

34 

27 

522424 

596 

974431 

74 

547943 

670 

452057 

33 

28 

522781 

596 

974436 

74 

548345 

670 

451655 

32 

29 

523138 

595 

974391 

74 

548747 

669 

451253 

31 

30 

523495 

595 

974347 

75 

549149 

669 

450851 

30 

31 

9.523852 

594 

9.974302 

75 

9.549550 

668 

10.450450 

29 

32 

524208 

594 

974-257 

75 

549951 

668 

450049 

28 

33 

524564 

593 

974212 

75 

550352 

667 

449648 

27 

34 

524920 

593 

974167 

75 

550752 

667 

449248 

26 

35 

525275 

592 

974122 

75 

551152 

666 

448848 

25 

36 

525630 

591 

974077 

75 

551552 

666 

448448 

24 

37 

525984 

591 

974032 

75 

55J953 

665 

448048 

23 

38 

536339 

590' 

973987 

75 

552351 

665 

447649 

22 

39 

52H093 

590 

973942 

75 

552750 

665 

447250 

21 

40 

527046 

589 

973897 

75 

553149 

664 

446851 

20 

41 

9.527400 

589 

9.973852 

75 

9.553548 

664 

10.446452 

19 

42 

527753 

588 

973807 

75 

553946 

663 

446054 

18 

43 

528105 

588 

973701 

75 

554344 

663 

445656 

17 

44 

528458 

587 

973716 

76 

554741 

662 

445259 

16 

45 

528810 

587 

973671 

7(5 

555138 

662 

444861 

15  1 

46 

529161 

586 

973625 

76 

555536 

661 

444464 

14  , 

47 

529513 

586 

973580 

76 

555933 

661 

444067 

13 

48 

539864 

585 

973535 

76 

556329 

660 

443671 

12 

49 

530215 

585 

973489 

76 

556725 

660 

443275 

11 

50 

530565 

584 

973444 

76 

557121 

659 

442879 

10 

51 

9.530915 

584 

9.973398 

76 

9.557517 

659 

10.442483 

9 

52 

531265 

583 

973352 

76 

557913 

659 

442087 

8 

53 

531614 

582 

973307 

76 

558308 

658 

441692 

7 

54 

531903 

582 

973261 

76 

558702 

658 

441298 

6 

55 

532312 

581 

973215 

76 

559097 

657 

440903 

5 

5(5 

532661 

581 

973169 

76 

559491 

657 

440509 

4 

57 

KWOOH 

580 

973124 

76 

559885 

656 

440115 

3 

58 

533357 

580 

973078 

76 

560279 

656 

439721 

2 

59 

533704 

579 

973032 

77 

560673 

655 

439327 

1 

60 

534052 

578 

972986 

77 

561066 

655 

438934 

0 

|  Cosine  |      |   Sine   |    |  Cotang.  |     _    Tang.   |  M. 

70  Degrees. 


Degrees.)    A  TABLE  OF  LOGARITHMIC 


M.    Sine     D.     Cosine  |  D.  |   Tang.   |   D.     Cotang.  | 

0 

9.534052 

578 

9.97293S 

77 

9.551066 

655 

10.4  89  4 

60 

1 

534399 

577 

972940 

77 

5(51459 

654 

438341 

59 

2 

534745 

577 

972394 

77 

5618.")! 

654 

438149 

58 

3 

535092 

577 

97-2348 

77 

562244 

653 

437756 

57 

4 

535438 

576 

97-2302 

77 

502636 

653 

437364 

56 

5 

535783 

576 

972755 

77 

563028 

653 

'  43(5972 

55 

6 

533129 

575 

972709 

77 

533419 

652 

43:>531 

54 

7 

53(5474 

574 

972663 

77 

563811 

652 

436189 

53 

8 

536818 

574 

972617 

77 

5(542)2 

651 

435798 

52 

9 

537163 

573 

972570 

77 

564592 

651 

435408 

51 

10 

5375J7 

573 

972524 

77 

5154933 

650 

435017 

50 

11 

9.537851 

572 

9.972478 

77 

9.565373 

650 

10.434627 

49 

1-2 

538194 

572 

972431 

78 

565763 

619 

434237 

48 

13 

838538 

571 

972335 

78 

566153 

649 

433847 

47 

14 

538880 

571 

972338 

73 

556542 

649 

433458 

46 

15 

539223 

570 

9722.)  1 

78 

566932 

648 

433068 

45 

16 

5395155 

570 

972345 

78 

567320 

648 

4326  Si) 

44 

17 

539907 

569 

972198 

78 

567709 

647 

432291 

43 

18 

540-249 

569 

972151 

78 

563093 

647 

431902 

42 

19 

540590 

568 

972105 

78 

563488 

646 

431514 

41 

20 

54J931 

568 

972058 

78 

563373 

646 

431127 

40 

21 

9.541272 

567 

9.972011 

78 

9.5*59261 

645 

10.430739 

39 

22 

541613 

567 

971914 

73 

569(548 

645 

430352 

33 

23 

541953 

566 

971917 

78 

570035 

645 

4299-55 

37 

24 

542293 

566 

971870 

78 

570422 

644 

429578 

36 

25 

542832 

565 

971823 

/8 

570309 

644 

421)191 

35 

26 

542971 

565 

971776 

78 

571195 

643 

4288J5 

34 

27 

543310 

584 

971729 

79 

571.581 

643 

428419 

33 

28 

543!549 

564 

971682 

79 

571967 

642 

428033 

32 

29 

543937 

5!53 

971635 

79 

57-2352 

642 

427648 

31 

30 

514325 

563 

971583 

79 

572733 

642 

427252 

30 

31 

9.544663 

5(52 

9.971540 

79 

8.  573123 

641 

10.426877 

29 

32 

5450JO 

562 

971493 

79 

573507 

641 

42!>493 

28 

33 

545338 

561 

97144!! 

79 

573893 

640 

4-2(5108 

27 

34 

545674 

561 

971398 

79 

574276 

640 

425724 

26 

35 

54(5011 

580 

971351 

79 

574560 

639 

425340 

25 

36 

516347 

5f50 

9713J3 

79 

575044 

639 

424956 

24 

37 

546683 

559 

971256 

79 

575427 

639 

424573 

23 

38 

547019 

559 

97  J  208 

79 

5758JO 

638 

424193 

22 

39 

547354 

553 

9711(51 

79 

576193 

638 

423807 

21 

40 

547689 

553 

971113 

79 

576576 

637 

423424 

20 

41 

9.548024 

557 

9.971066 

80 

9.576958 

637 

10.423041 

19 

42 

548359 

557 

971018 

88 

577341 

636 

422659 

18 

43 

5481593 

555 

970970 

80 

577723 

636 

422277 

17 

44 

549027 

556 

970922 

80 

578104 

636 

421896 

]6 

45 

549360 

555 

970874 

80 

578486 

635 

421514 

15 

46 

549593 

555 

970827 

8D 

578887 

635 

421133 

14 

47 

550026 

554 

970779 

80 

579248 

634 

420752 

13 

48 

550359 

554 

970731 

80 

579829 

634 

420371 

12 

49 

550692 

553 

970683 

80 

580009 

634 

419991 

11 

50 

551024 

553 

970635 

80 

580389 

633 

419811 

10 

51 

9  551356 

552 

9.970586 

80 

9.580769 

633 

10.419231 

9 

52 

551687 

552 

970538 

80 

581149 

632 

418851 

8 

53 

552018 

552 

970490 

83 

581528 

632 

413472 

7 

54 

552349 

551 

970442 

80 

581907 

632 

418093 

6 

55 

552680 

551 

970394 

80 

582286 

631 

417714 

5 

58 

553010 

550 

970345 

81 

582665 

631 

417335 

4 

57 

553341 

550 

970297 

81 

583043 

630 

416957 

3 

58 

553670 

549 

970249 

81 

583422 

630 

416578 

2 

59 

554000 

549 

970230 

81 

583800 

629 

416200 

1 

CO 

554329 

548 

970152 

81 

584177 

629 

415823 

0 

Cosine  |         Sine   |    |  Cotang.       |   Tang.   |  M. 

69  Degrees 


SINES  AND  TANGENTS.     (21  Degrees.) 


39 


M.    Sine     D.   |  Cosine  |  D.  |   Tang.     D.     Cotang.  | 

0 

9.5J43-29 

548 

9.970152 

81 

9.584177 

629 

10.415823 

60 

1 

5541)58 

548 

970103 

81 

f>->  i  :,:>.-> 

62J 

415445 

59 

2 

554987 

547 

9700:>3 

81 

584932 

628 

415068 

58 

3 

555315 

547 

970U06 

81 

585309 

628 

414691 

57 

4 

555643 

546 

969957 

81 

585686 

627 

414314 

56 

5 

555971 

546 

959909 

81 

586062 

627 

413938 

55 

6 

556299 

545 

969860 

81 

586439 

627 

413561 

54 

7 

55T.626 

545 

989811 

81 

586815 

626 

413185 

53 

8 

556953 

544 

969762 

81 

587190 

626 

412810 

52 

9 

557-280 

544 

969714 

81 

587566 

625 

412434 

51 

10 

557606 

543 

969665 

81 

587941 

625 

412059 

50 

11 

9.557OT2 

543 

9.969616 

82 

9.588316 

625 

10.411684 

49 

12 

558-258 

543 

969567 

82 

588691 

624 

411309 

48 

13 

558583 

542 

969518 

82 

589086 

624 

410934 

47 

14 

558909 

542 

969469 

82 

589440 

623 

410560 

46 

15 

55<h234 

541 

969420 

82 

589814 

623 

410186 

45 

16 

559558 

541 

969370 

82 

590188 

623 

409812 

44 

17 

559883 

540 

969321 

82 

590562 

622 

409438 

43 

18 

560207 

540 

969272 

82 

591)935 

622 

409065 

42 

19 

560531 

539 

969223 

82 

591308 

622 

408692 

41 

21) 

560855 

539 

969173 

82 

591681 

621 

408319 

40 

21 

9.561178 

538 

9.969124 

82 

9.592054 

621 

10.407946 

39 

22 

561501 

538 

969075 

82 

592426 

620 

407574 

38 

23 

561824 

537 

969025 

82 

592798 

620 

407202 

37 

24 

562146 

537 

968976 

82 

593170 

619 

406829 

36 

25 

562468 

536 

968926 

83 

593542 

619 

406458 

35 

26 

562790 

536 

968877 

83 

593914 

618 

406086 

34 

27 

563112 

536 

968827 

83 

594285 

618 

405715 

33 

28 

563433 

535 

988777 

83 

594656 

618 

405344 

32 

29 

563755 

535 

968728 

83 

595027 

617 

404973 

31 

30 

564075 

534 

968678 

83 

595398 

617 

404602 

30 

31 

9.564396 

534 

9.968628 

83 

9.595768 

617 

10.404232 

29 

32 

564716 

533 

968578 

83 

596138 

616 

403862 

28 

33 

565036 

533 

968528 

83 

596508 

616 

403492 

27 

34 

565356 

532 

968479 

83 

596878 

616 

403122 

26 

35 

565676 

532 

968429 

83 

597247 

615 

402753 

25 

36 

565995 

531 

968379 

83 

597616 

615 

402384 

24 

37 

566314 

531 

9(38329 

83 

597985 

615 

402015 

23 

38 

566632 

531 

068378 

83 

598354 

614 

401646 

22 

39 

566951 

530 

968228 

84 

598722 

614 

401278 

21 

40 

567269 

530 

968178 

84 

599091 

613 

400909 

20 

41 

9.567587 

529 

9.968128 

84 

9.599459 

613 

10.400541 

19 

42 

587904 

529 

968078 

84 

599827 

613 

400173 

18 

43 

568222 

528 

968027 

84 

600194 

612 

399806 

17 

44 

568539 

528 

967977 

84 

600562 

612 

399438 

16 

45 

568856 

528 

967927 

84 

600929 

611 

399071 

15 

46 

569172 

527 

967876 

84 

601296 

611 

398704 

14 

47 

569488 

527 

957826 

84 

601662 

611 

398338 

13 

48 

569804 

526 

967775 

84 

602029 

610 

397971 

12 

49 

570120 

526 

967725 

84 

602395 

610 

397605 

11 

50 

570435 

525 

967674 

84 

602761 

610 

397239 

10 

51 

9.570751 

525 

9.987624 

84 

9.603127 

609 

10.396873 

9 

52 

571066 

524 

957573 

84 

603493 

609 

396507 

8 

53 

571380 

524 

967522 

85 

603858 

609 

396142 

7 

54 

571695 

523 

967471 

85 

604223 

608 

395777 

6 

55 

572009 

523 

967421 

85 

604588 

608 

395412 

5 

56 

5723-23 

523 

9-77370 

85 

604953 

607 

395047 

4 

57 

572636 

522 

967319 

85 

605317 

607 

394683 

3 

53 

572950 

5-22 

967268 

85 

605682 

607 

394318 

I 

59 

573263 

521 

987217 

85 

606046 

606 

393!).>4 

1 

60 

573575 

521 

967168 

85 

61)6410 

606 

393590 

0 

Cosine  |      |   Sine   |    |  Cotang.       |   Tang.   |  M.  ) 

Degrees. 


40 


Degrees.)    A  TABI.K  OP  LOGARITHMIC 


M.    Sine   |   D.   |  Cosine  |  D.    Tang.   |   D.     Cotang  | 

0 

9.573575 

521 

9.967166   85   9.600410 

606 

10.393590 

60 

1 

573888 

520 

967115 

85 

606773 

606 

393227 

59 

2 

574200 

520 

967064 

85 

607137 

605 

392863 

58 

3 

574512 

519 

967013 

85 

607500 

605 

392500 

57 

4 

57-18-24 

519 

966961 

85 

607863 

604 

392137 

56 

5 

575136 

519 

966910 

85 

608225 

604 

391775 

55 

6 

575447 

518 

966859 

85 

608588 

604 

391412 

54 

7 

575758 

518 

9G68U8 

85 

608950 

603 

391050 

53 

8 

576069 

517 

966756 

86 

609312 

603 

390688 

52 

9 

576379 

517 

966705 

86 

609674 

603 

390326 

51 

10 

576689 

516 

966653 

86 

610036 

602 

389964 

50 

11 

9.576999 

516 

9.966602 

86 

9.61&397 

602 

10.389603 

49 

12 

577309 

516 

966550 

86 

610759 

602 

389241 

48 

13 

577618 

515 

966499 

86 

611120 

601 

388880 

47 

14 

577927 

515 

966447 

86 

61  J  480 

601 

388520 

46 

15 

578236 

514 

966395 

86 

611841 

601 

388  J  59 

45 

16 

578545 

514 

966344 

86 

612201 

600 

387799 

44 

17 

578853 

513 

966292 

86 

612561 

600 

387439 

43 

18 

579162 

513 

966240 

86 

612921 

600 

387079 

42 

19 

579470 

513 

966188 

86 

613281 

599 

386719 

41 

20 

579777 

512 

966136 

86 

613641 

599 

380359 

40 

21 

9.580085 

512 

9.966085 

87 

9.614000 

598 

10.386000 

39 

-22 

580392 

511 

966033 

87 

614359 

598 

385641 

38 

23 

580699 

511 

965981 

87 

614718 

598 

385282 

37 

24 

581005 

511 

965928 

87 

615077 

597 

384923 

36 

25 

581312 

510 

965876 

87 

615435 

597 

384565 

35 

26 

581618 

510 

965824 

87 

615793 

597 

384207 

34 

27 

581924 

509 

965772 

87 

616151 

596 

383849 

33 

28 

5822-29 

509 

965720 

87 

616509 

596 

383491 

32 

29 

582535 

509 

965668 

87 

616867 

596 

383133 

31 

30 

582840 

508 

965615 

87 

617224 

595 

382776 

30 

31 

9.583145 

508 

9.965563 

87 

9.617582 

595 

10.382418 

29 

32 

583449 

507 

965511 

87 

617939 

595 

382061 

28 

33 

583754 

507 

965458 

87 

6J8295 

594 

381705 

27 

34 

584058 

506 

965406 

87 

618652 

594 

381348 

26 

35 

584361 

506 

965353 

88 

619008 

594 

380992 

25 

36 

584665 

506 

965301 

88 

619364 

593 

380636 

24 

37 

584968 

505 

965248 

88 

619721 

593 

380279 

23 

38 

585272 

505 

965195 

88 

620076 

593 

379924 

22 

39 

585574 

504 

965143 

88 

620432 

59:2 

379568 

21 

40 

585877 

504 

965090 

88 

620787 

592 

379213 

20 

41 

9.586179 

503 

9.965037 

88 

9.621142 

592 

10.378858 

19 

42 

586482 

503 

964984 

88 

621497 

591 

378503 

18 

43 

586783 

503 

964931 

88 

621852 

591 

378148 

17 

44 

587085 

502 

964879 

88 

62-2-207 

590 

377793 

16 

45 

587386 

502 

964826 

88 

622561 

590 

377439 

15 

46 

587688 

501 

964773 

88 

62-2915 

590 

377085 

14 

47 

587989 

501 

9(54719 

88 

623269 

589 

376731 

13 

48 

588289 

501 

964666 

89 

6-23623 

589 

37(5377 

12 

49 

588590 

500 

964613 

89 

623976 

589 

37f'024 

11 

50 

588890 

500 

964560 

89 

624330 

588 

375670 

10 

51 

9.589190 

499 

9.964507 

89 

9.624C83 

588 

10.375317 

9 

52 

589489 

499 

964454 

89 

625036 

588 

3749(54 

8 

53 

589789 

499 

964400 

89 

625388 

587 

374612 

7 

54 

590088 

498 

964347 

89 

625741 

587 

374259 

6 

55 

590387 

498 

964294 

89 

626093 

587 

373907 

5 

56 

593686 

497 

964940 

89 

62(5445 

586 

373555 

4 

57 

590984 

497 

964187 

89 

tvjt;-;i7 

586 

373203 

3 

58 

591282 

497 

9B4133 

89 

6-27149 

586 

372851 

2 

59 

591580 

496 

964080 

89 

627501 

585 

372499 

1 

60 

59187Q 

496 

9W026 

89 

627852 

585 

372148 

0 

Coeine  |      |   Sine   |     Cotang.        |   Tang,   j  M.  ! 

67  Degrees. 


SINES  AND  TANGENTS.     (23  Degrees.) 


4! 


M.  |   Sine   |   D.   |   Cosine   D.  |   Tang.  |   D.  |  Cotang.  | 

0 

9.591878 

496 

0.984036 

89 

9.6278:12 

585 

10.372148 

C,u 

1 

59-2170 

495 

9(53972 

89 

628203 

585 

3717!)7 

59 

2 

59-2473 

495 

,  963919 

89 

628554 

585 

371446 

58 

3 

592770 

495 

983865 

90 

628905 

584 

371095 

57 

4 

593067 

494 

9153811 

90 

629255 

584 

370745 

56 

5 

593303 

494 

;Mi:i7^7 

90 

fi29;50« 

583 

370394 

55 

6 

5t3659 

493 

963704 

90 

629956 

583 

370044 

54 

7 

593:>5.1 

493 

963650 

90 

630306 

583 

369694 

53 

8 

59425J 

493 

9(53596 

90 

630656 

583 

369344 

52 

9 

594547 

492 

963542 

90 

631005 

582 

368995 

51 

10 

594842 

492 

963488 

90 

631355 

582 

368645 

50 

11 

9.595137 

491 

9.963434 

90 

9.631704 

582 

10.368296 

49 

12 

595432 

491 

963379 

93 

632053 

581 

367947 

48 

13 

595727 

491 

963325 

90 

632401 

581 

367599 

47 

14 

59602] 

490 

903271 

90 

632750 

581 

367250 

46 

15 

508315 

490 

963217 

90 

633098 

580 

366902 

45 

16 

59:5609 

489 

963163 

9D 

633447 

580 

366553 

44 

17 

590903 

489 

963108 

91 

633795 

580 

366205 

43 

18 

51)7191} 

489 

963054 

91 

634143 

579 

365857 

42 

lit 

597490 

488 

962999 

91 

634490 

579 

365510 

41 

20 

597783 

488 

962945 

91 

634838 

579 

365162 

40 

21 

9.598075 

487 

9.962390 

91 

9-635185 

578 

10.364815 

39 

22 

598368 

487 

962836 

91 

635532 

578 

364468 

38 

23 

598600 

487 

962781 

91 

635879 

578 

364121 

37 

24 

538952 

486 

962727 

91 

636226 

577 

363774 

36 

25 

599244 

486 

962672 

91 

636572 

577 

36342S 

35 

86 

599530 

485 

962617 

91 

636919 

577 

363081 

34 

27 

5998-27 

485 

962562 

91 

637265 

577 

362735 

33 

28 

61)0118 

485 

962508 

91 

637611 

576 

382389 

32 

29 

600409 

484 

962153 

91 

637956 

576 

362044 

31 

30 

61)0700 

484 

9ii23J8 

92 

638302 

576 

361698 

30 

31 

9.600990 

484 

9.962343 

92 

9.638647 

575 

10.351353 

29 

32 

601280 

483 

962288 

92 

638992 

575 

361008 

28' 

33 

601570 

483 

9(52233 

92 

639337 

575 

360663 

27 

34 

601880 

482 

962178 

92 

639682 

574 

360318 

26 

35 

602150 

482 

962123 

92 

640027 

574 

359973 

25 

M 

602439 

482 

983067 

92 

640371 

574 

359629 

24 

37 

602728 

481 

982012 

92 

610716 

573 

359284 

23 

38 

603017 

481 

981957 

92 

641000 

573 

358940 

22 

39 

60331)5 

481 

951902 

92 

641404 

573 

358596 

21 

;  •»" 

603594 

480 

961846 

93 

641747 

572 

358253 

20 

41 

9.603882 

480 

9.961791 

92 

9-642091 

572 

10.357909 

19 

42 

604170 

479 

9:51735 

92 

642434 

572 

35750(5 

18 

43 

604457 

479 

961680 

92 

642777 

572 

357223 

17 

44 

604745 

479 

901624 

93 

643120 

571 

356880 

16 

45 

605032 

478 

961569 

93 

643463 

571 

356537 

15 

4(5 

605319 

478 

961513 

93 

64381)0 

571 

356194 

14 

v47 

605(503 

478 

961458 

93 

644148 

570 

355852 

13 

IB 

605892 

477 

93]  402 

93 

644490 

570 

355510 

12 

49 

606179 

477 

981346 

93 

644832 

570 

355168 

11 

50 

606465 

476 

981299 

93 

645174 

569 

354826 

10 

51 

9.696751 

476 

9.981235 

93 

9.645516 

569 

10.354484 

9 

-)•_' 

607038 

476 

961179 

93 

645857 

569 

3.141  13 

8 

so 

607322 

475 

901123 

93 

640199 

569 

353801 

7 

54 

607607 

475 

981067 

93 

646540 

568 

3534(50 

6 

55 

607892 

474 

981011 

93 

646881 

568 

353119 

5 

5ti 

808177 

474 

900955 

93 

647222 

568 

352778 

4 

57 

608461 

474 

980899 

93 

647562 

567 

352438 

3 

58 

608745 

473 

900813 

94 

647903 

567 

352097 

2 

59 

609029 

473 

960786 

91 

648243 

567 

351757 

1 

60 

609313 

473 

960730 

94 

MKJki 

-«»(• 

351417 

0 

|  Cosine  |      |   Sine      |  Cotang.  |         Tang.  |  M.  1 

42 


(24  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.  |   Sine  |   D    Cosine   j  D.  |  Tang   |   D.   |  Cotang.  | 

0 

9.609313 

473 

9.960730 

94 

9.648583 

566 

10.351417 

GO  1 

1 

C09597 

472 

960674 

M 

648923 

566 

351077 

59 

2 

609880 

472 

960618 

94 

649263 

566 

350737 

58 

3 

610164 

472 

960561 

94 

649602 

568 

350398 

57 

4 

610447 

471 

960505 

94 

649942 

565 

350058 

56 

5 

610729 

471 

9G0448 

94 

650281 

565 

349719 

55 

6 

611012 

470- 

960332 

94 

650620 

565 

349380 

54 

7 

6J  1294 

470 

960335 

94 

650959 

564 

349041 

53  1 

8 

611576 

470 

960279 

94 

651297 

564 

648703 

52 

9 

611858 

469 

960222 

94 

651636 

564 

348364 

51 

10 

612140 

469 

9G01G5 

94 

G51974 

563 

318026 

50 

11 

9.612421 

469 

9.960109 

95 

9.652312 

563 

10.347688 

49 

12 

612702 

468 

960052 

95 

052650 

563 

347350 

4d 

13 

612983 

468 

959995 

95 

652988 

563 

347012 

47 

14 

613264 

467 

959938 

95 

6533-26 

562 

34R674 

46 

15 

613545 

467 

959882 

95 

653663 

562 

346337 

45 

16 

613825 

467 

959825 

95 

654000 

562 

341.000 

44 

17 

614105 

466 

959768 

95 

654337 

561 

345663 

43 

18 

614385 

466 

9597J1 

95 

654674 

561 

345326 

42 

19 

614C65 

466 

959654 

95 

655011 

561 

344989 

41 

20 

614944 

465 

959506 

95 

655348 

561 

344652 

40 

21 

9.615223 

465 

9.959539 

95 

9.655684 

560 

10.344316 

39 

22 

615502 

465 

959482 

95 

656020 

560 

343980 

38 

23 

615781 

464 

959425 

95 

656356 

560 

343644 

37 

24 

616060 

464 

9593(58 

95 

656692 

559 

343308 

36  | 

25 

616338 

464 

959310 

96 

657028 

559 

342972 

35 

26 

61(1616 

463 

959253 

96 

657364 

559 

342636 

34 

27 

616894 

463 

959195 

96 

6571)99 

559 

342201 

33 

28 

617172 

462 

959138 

96 

658034 

558 

341960 

32 

29 

617450 

462 

959081 

96 

658369 

558 

341631 

31 

30 

617727 

462 

959023 

96 

658704 

558 

341296 

30 

31 

9.618004 

461 

9.958965 

90 

9.659039 

558 

10.  340901 

29 

32 

618381 

461 

9o8908 

96 

659373 

557 

340G-27 

28 

33 

618558 

461 

958850 

96 

659708 

557 

340292 

27 

34 

618834 

460 

958792 

96 

660042 

557 

3399.58 

26 

35 

619110 

460 

958734 

95 

660376 

557 

339624 

25 

36 

619386 

460 

958677 

96 

660710 

556 

339290 

24 

37 

619662 

459 

958G19 

96 

661043 

556 

338957 

23 

38 

619938 

459 

958561 

96 

661377 

556 

338623 

22 

39 

620213 

459 

9.58503 

97 

661710 

555 

338290 

21 

40 

620488 

458 

958445 

97 

662043 

555 

337957 

20 

41 

9.6207G3 

458 

9.958387 

97 

9.662376 

555 

10.337624 

19 

42 

621038 

457 

958329 

97 

662709 

554 

337291 

18 

43 

621313 

457 

958271 

97 

663042 

554 

336958 

17 

44 

621587 

457 

958213 

97 

663375 

554 

336625 

16 

45 

6218G1 

456 

958154 

97 

663707 

554 

336293 

15 

46 

622135 

456 

958096 

97 

664039 

553 

335961 

14 

47 

622409 

456 

958038 

97 

664371 

553 

335629 

13 

48 

622682 

455 

957979 

97 

664703 

553 

335297 

12 

49 

622936 

455 

957921 

97 

665035 

553 

334965 

11 

50 

623229 

455 

957863 

97 

665366 

552 

334634 

10 

51 

9.623502 

454 

9.957804 

97 

9.665697 

552 

10.334303 

9 

52 

623774 

454 

957746 

98 

66G029 

552 

333971 

8 

53 

621047 

454 

957087 

98 

666360 

551 

333640 

7 

54 

624319 

453 

957628 

98 

666691 

551 

333309 

6 

55 

624591 

453 

957570 

98 

667021 

551 

332979 

5 

56 

G248f>3 

453 

957511 

98 

667352 

551 

332648 

4 

57 

623135 

452 

957452 

98  ' 

667682 

550 

332318 

3 

58 

625406 

452 

957393 

98 

6f8:)13 

550 

331987 

0 

'  59 

625(777 

452 

957335 

9H 

668343 

550 

331657 

1 

60 

625948 

451 

95727G 

98 

668372 

550 

331328 

0 

1  Cosine  |      |   Sine   |      Cotang.  |      |   Tang.  |  M. 

Go  Degrees. 


SINES  AND  TANGENTS.     (25  Degrees.) 


j  M.     Sine    D.    Cosine    D.   |   Tang.     D.     CoLing.  | 

0 

9.625948 

451 

9.957276 

98   U.  668673 

550 

10.331327 

6( 

1 

626219 

451 

957217 

98 

669002 

549 

330998 

5« 

2 

626490 

451 

957158 

98 

(5f!93:J2 

549 

330668 

5£ 

3 

6207(50 

450 

957099 

98 

669(561 

549 

330339 

57 

4 

627030 

450 

957040 

98 

669991 

548 

330009 

56 

5 

637300 

450 

956981 

98 

670320 

548 

329080 

55 

6 

637570 

449 

956921 

99 

670049 

548 

329351 

54 

7 

627840 

449 

956843 

99 

070977 

548 

329023 

51 

8 

628109 

449 

956803 

99 

671306 

547 

328694 

5£ 

9 

628:578 

448 

950744 

99 

671634 

547 

328366 

5] 

10 

628647 

448 

956684 

99 

671963 

547 

328037 

50 

11 

9.628916 

447 

9.956625 

99 

9.672291 

547 

10.327709 

4:1  : 

12 

629185 

447 

956506 

99 

672619 

546 

327381 

48 

13 

629453 

447 

950506 

99 

672947 

546 

327053 

47 

14 

6297-21 

446 

956447 

99 

673274 

546 

326726 

46 

15 

629989 

446 

956387 

99 

673602 

546 

326398 

45 

16 

630257 

446 

956327 

99 

673929 

545 

326071 

44 

17 

630.124 

446 

9562(58 

99 

674257 

545 

325743 

43 

18 

630792 

445 

956208 

100 

674584 

545 

325416 

42 

M 

631051) 

445 

956148 

100 

674910 

544 

325090 

41 

20 

631326 

445 

956089 

100 

675237 

544 

324763 

40 

21 

9.631593 

444 

9.956029 

100 

9.675564 

544 

10.324436 

39 

22 

631859 

444 

955969 

100 

675890 

544 

324110 

38 

23 

632125 

444 

955909 

100 

676216 

543 

323784 

37 

24 

632392 

443 

955849 

100 

676543 

543 

323457 

36 

25 

632658 

443 

955789 

100 

676869 

543 

323131 

35 

26 

632923 

443 

955729 

100 

677194 

543 

3i2806 

34 

27 

633189 

442 

955669 

100 

677520 

542 

322480 

33 

28 

633454 

442 

955609 

100 

677846 

542 

322154 

32 

29 

633719 

442 

955548 

100 

678171 

542 

321829 

31 

30 

633984 

441 

955488 

100 

67iM96 

542 

321504 

30 

31 

9.634249 

441 

9.955428 

101 

9.678821 

541 

10.321179 

29 

32 

634514 

440 

955368 

101 

679146 

541 

320854 

28 

33 

634778 

440 

955307 

101 

679471 

541 

320529 

27 

34 

635042 

440 

955247 

101 

679795 

541 

320205 

26 

35 

635306 

439 

955186 

101 

680120 

540 

319880 

25 

36 

635570 

439 

955126 

101 

680444 

540 

319556 

24 

37 

635834 

439 

955065 

101 

680768 

540 

319232 

23 

38 

636097 

438 

955005 

101 

681092 

540 

318908 

22 

39 

636360 

438 

954944 

101 

681416 

539 

318584 

21 

40 

636623 

438 

954883 

101 

681740 

539 

318260 

20 

41 

9.636886 

437 

9.954823 

101 

9.682063 

539 

10.317937 

19 

42 

637148 

437 

954762 

101 

682387 

539 

317613 

18 

43 

637411 

437 

954701 

101 

682710 

538 

317290 

17 

44 

637673 

437 

954640 

101 

683033 

538 

316967 

16 

45 

637935 

436 

954579 

101 

683356 

538 

316644 

15 

46 

638197 

436 

954518 

102 

683679 

538 

316321 

14 

47 

638458 

436 

954457 

102 

684001 

537 

315999 

13 

48 

638720 

435 

954396 

102 

684324 

537 

315676 

12 

49 

638981 

435 

954335 

102 

684646 

537 

315354 

11 

50 

639242 

435 

954274 

102 

684968 

537 

315032 

10 

51 

9.639503 

434 

9.954213 

102 

9.685290 

536 

10.314710 

9 

52 

639764 

434 

954152 

102 

685672 

536 

314388 

8 

53 

640024 

434 

95-1090 

102 

685934 

536 

314066 

7 

54 

640284 

433 

954029 

102 

686255 

536 

313745 

6 

55 

640544 

433 

953908 

102 

686577 

535 

313423 

5 

56 

640804 

433 

953906 

102 

686898 

535 

313102 

4 

57 

641064 

432 

953845 

102 

087*19 

535 

312781 

3 

58 

641324 

432 

953783 

102 

687540 

535 

312460 

2 

59 

641584 

432 

953722 

103 

687861 

534 

312139 

1 

60 

641842 

431 

953660 

103 

r,Krflr2 

534 

3I1H1H 

o  I 

|  Cosine  |        Sine   |        Cotang.          Tang.   M. 

64  Degrees. 


44 


(26  Degrees.)    A  TABLE  OF  LOGARITHMIC 


'  M.  |   Sine   |   D.  |  Cosine  |  D.  |   Tang.  |   D.  |  Cotang. 

0 

9.641842 

431 

9-953660 

103 

9.688182 

534 

10.311818 

60  I 

1 

642101 

431 

953599 

103 

088502 

534 

311498 

59 

2 

642360 

431 

953537 

103 

688823 

534 

311177 

58 

3 

642618 

430 

953475 

103 

689143 

533 

310857 

57 

4 

642877 

430 

953413 

103 

689463 

533 

310537 

56 

5 

643135 

430 

953352 

103 

689783 

533 

310217 

55 

6 

643393 

430 

953290 

103 

690103 

533 

309897 

54 

7 

643050 

429 

953228 

103 

690423 

533 

309577 

53 

8 

643908 

429 

953166 

103 

690742 

532 

309258 

52 

9 

644165 

429 

953104 

103 

691062 

532 

308938 

51 

10 

644423 

428 

953042 

103 

691381 

532 

308619 

50 

11 

9.644680 

428 

9.952980 

104 

9.691700 

531 

10.308300 

49 

12 

644936 

428 

952918 

104 

692019 

531 

307981 

48 

13 

645193 

427 

952855 

104 

692338 

531 

307662 

47 

14 

645450 

427 

952793 

104 

692656 

531 

307344 

46 

15 

645706 

427 

952731 

104 

692975 

531 

307025 

45 

16 

645962 

426 

952669 

104 

693293 

530 

306707 

44 

17 

646218 

426 

952606 

104 

693612 

530 

308388 

43 

18 

646474 

426 

952544 

104 

693930 

530 

306070 

42 

19 

646729 

425 

952481 

104 

694248 

530 

305752 

41 

20 

646984 

425 

952419 

104 

694566 

529 

305434 

40 

21 

9.647240 

425 

9.952356 

104 

9.694883 

529 

10.305117 

39 

22 

647494 

424 

952294 

104 

695201 

529 

304799 

38 

23 

647749 

424 

952231 

104 

695518 

529 

304482 

37 

24 

648004 

424 

952168 

105 

695836 

529 

304164 

36 

25 

648258 

424 

952106 

105 

696153 

528 

303847 

35 

26 

648512 

423 

952043 

105 

696470 

528 

303530 

34 

27 

648766 

423 

951980 

105 

696787 

528 

303213 

33 

28 

649020 

423 

951917 

105 

697103 

528 

302897 

32 

29 

649274 

422 

951854 

105 

697420 

527 

302580 

31 

30 

649527 

422 

951791 

105 

697736 

527 

302264 

30 

31 

9-649781 

422 

9.951728 

105 

9.693053 

527 

10.301947 

29 

32 

650034 

422 

951665 

105 

693369 

527 

301631 

28 

33 

650287 

421 

951602 

105 

698685 

526 

301315 

27 

34 

650539 

421 

951539 

105 

699001 

526 

300999 

26 

35 

650792 

421 

951476 

105 

699316 

526 

300684 

25 

36 

651044 

420 

951412 

105 

699632 

526 

300368 

24 

37 

651297 

420 

951349 

106 

699947 

526 

3001)53 

23 

38 

651549 

420 

951286 

106 

700263 

525 

299737 

22 

39 

651800 

419 

951222 

106 

700578 

525 

299422 

21 

40 

652052 

419 

951159 

106 

700893 

525 

299107 

20 

41 

9.652304 

419 

9.951096 

106 

9.701208 

524 

10.298792 

19 

42 

652555 

418 

951032 

106 

701523 

524 

298477 

18 

43 

652806 

418 

950968 

106 

701837 

524 

298163 

17 

44 
45 

653057 
653308 

418 
418 

950905 
950841 

106 
106 

702152 
702466 

524 
524 

297848 
297534 

16 
15 

46 

653558 

417 

950778 

106 

702780 

523 

297220 

14 

47 

653808 

417 

950714 

106 

703095 

523 

296905 

13 

48 

654059 

417 

950650 

106 

703409 

523 

296591 

12 

49 

654309 

416 

950586 

106 

703723 

523 

296277 

11 

50 

654558 

416 

950522 

107 

704036 

522 

295964 

10 

51 

9.654808 

416 

9.950458 

107 

9.704350 

522 

10.295650 

9 

52 

655058 

416 

950394 

107 

704663 

522 

295337 

8 

53 

655307 

415 

950330 

107 

704977 

522 

295023 

7 

54 

655556 

415 

950266 

107 

705290 

522 

294710 

6 

55 

655805 

415 

950202 

107 

705603 

521 

294397 

5 

56 

656054 

414 

950138 

107 

705916 

521 

294084 

4 

57 

656302 

414 

IS0074 

107 

706228 

521 

293772 

3 

58 

656551 

414 

950010 

107 

706541 

521 

293459 

2 

59 

656799 

413 

949945 

107 

706854 

521 

293146 

1 

60 

657047 

413 

949881 

107 

707166 

520 

292834 

0 

|  Cosine  |      |   Sine   |      Cotang.  | 

Tang.  |  M.  | 

63  Degrees. 


SINES  AND  TANGENTS.     (27  Degrees.) 


-15 


M.  |   Sine   |   D.  |   Cosine  |  D.    Tang.   |   D.    Cotonar.  1 

0 

9.657047 

413 

9.949881 

107 

9.  707  Kiii 

520 

10.2!>2H34 

60 

1 

657295 

413 

949816 

107 

707478 

520 

292522 

59 

2 

657542 

41-2 

949752 

107 

707790 

520 

292210 

58 

3 

657790 

412 

949688 

108 

708102 

520 

291898 

57 

4 

658037 

412 

949823 

108 

708414 

519 

291586 

56 

5 

658284 

419 

949558 

108 

708726 

519 

291274 

55 

6 

658531 

411 

949494 

108 

709037 

519 

291)963 

54 

7 

6r>>77H 

411 

949439 

108 

701)349 

519 

290651 

53 

8 

659025 

411 

9493154 

108 

709(560 

519 

290340 

52 

9 

(J5927I 

410 

.  949300 

108 

709971 

518 

290029 

51 

10 

659517 

410 

949235 

108 

710883 

518 

289718 

50 

11 

9.659763 

410 

9.949170 

108 

9.710593 

518 

10.289407 

49 

12 

6600:)9 

409 

949105 

108 

710904 

518 

289096 

48 

13 

<i<K)2.»5 

409 

949040 

108 

711215 

818 

288785 

47 

14 

660501 

409 

948975 

108 

711525 

517 

288475 

46 

15 

6JHI74G 

409 

948910 

108 

711836 

517 

288164 

45 

16 

660991 

408 

948845 

108 

712146 

517 

287854 

44 

17 

661236 

408 

948780 

109 

712456 

517 

287544 

43 

18 

661481 

408 

948715 

109 

71-27(5(5 

516 

287234 

42 

19 

661726 

407 

948650 

109 

713076 

516 

286924 

41 

20 

661970 

407 

948584 

109 

713386 

516 

286614 

40 

21 

9.662214 

407 

9.948519 

109 

9.71369S 

516 

10.286304 

39 

22 

662459 

407 

948454 

109 

714005 

516 

285995 

38 

23 

(5(52703 

406 

948388 

109 

714314 

515 

285686 

37 

24 

6G2946 

40(5 

948323 

109 

714624 

515 

285376 

36 

25 

663190 

406 

948257 

109 

714933 

515 

285067 

35 

20 

663433 

405 

948192 

109 

715242 

515 

284758 

34 

27 

(563677 

405 

9481215 

109 

715551 

514 

284449 

33 

28 

663920 

405 

948060 

109 

715800 

514 

234140 

32 

29 

664163 

405 

947995 

110 

716168 

514 

283832 

31 

30 

664406 

404 

947929 

110 

716477 

514 

283523 

30 

31 

9.664648 

404 

9.947863 

110 

9.716785 

514 

10.283215 

29 

32 

664891 

404 

947797 

110 

717093 

513 

282907 

28 

33 

665133 

403 

917731 

110 

717401 

513 

282599 

27 

34 

665375 

403 

947665 

110 

717709 

513 

282291 

26 

35 

6(55617 

403 

947C.OO 

110 

718017 

513 

281983 

25 

36 

665859 

402 

947533 

110 

718325 

513 

281675 

24 

37 

6G610i) 

402 

94";  4H7 

110 

718633 

512 

281367 

23 

38 

666342 

402 

947401 

110 

718940 

512 

281060 

22 

39 

668583 

-102 

947:j:{.-> 

110 

719248 

512 

280752 

21 

40 

666824 

401 

947289 

110 

719555 

512 

280445 

20 

41 

9.667065 

401 

9.947203 

110 

9.719862 

512 

10.280138 

19 

49 

667305 

401 

94713(5 

111 

720169 

511 

279831 

18 

43 

667546 

401 

947070 

111 

720476 

511 

279524 

17 

44 

667786 

400 

947004 

111 

720783 

511 

279217 

16 

45 

668027 

400 

946937 

111 

721089 

511 

278911 

15 

46 

668267 

400 

946871 

111 

721398 

511 

278604 

14 

47 

668508 

399 

946804 

111 

721702 

510 

278298 

13 

48 

668746 

399 

946738 

111 

722009 

510 

277991 

12 

49 

668936 

399 

946671 

111 

722315 

510 

277685 

11 

50 

669225 

399 

946604 

111 

722621 

510 

277379 

10 

51 

9.669464 

398 

9.946538 

111 

9.722927 

510 

10.277073 

9 

B 

669703 

398 

946471 

111 

723232 

509 

276768 

8 

53 

669942 

398 

946404 

111 

723538 

509 

276462 

7 

54 

670181 

397 

946337 

111 

723844 

509 

276156 

6 

55 

610419 

397 

946270 

112 

7-24149 

509 

275851 

5 

56 

670658 

397 

946203 

112 

724454 

509 

275546 

4 

57 

67089S 

397 

94(5136 

112 

724759 

508 

275241 

3 

58 

671134 

396 

946069 

112 

725065 

508 

274935 

2 

59    671372 

396 

946002 

112 

725369 

508 

274(531 

1 

60  1   671609 

396 

945935 

112 

725674 

508 

274326 

0 

|  Cosine  1         Sine   j     |  Cotang.  |      |   Tang.  |  M.  ' 

62  Degrees 


Degrees.)     A  TABLE  OT 


M.  )   Sine  I   D.   I  Cosine   f  D.  j  Tang.     D.   j  Cotang. 

0 

9.671009 

396 

9.1M5U:J5 

112 

9.725674 

508 

10.274326  1  00 

1 

671847 

395 

9458(18 

112 

725979 

508 

27402J 

59 

o 

67-2084 

395 

945800 

112 

7-J62C4 

507 

273718 

58 

3 

672321 

385 

945733 

112 

726583 

507 

273412 

57 

4 

672f>58 

395 

945666 

112 

7-ji;e.<i-. 

507 

27.1  JOS 

56 

5 

672795 

394 

9455JW 

1J2 

727197 

507 

272803 

55 

6 

673032 

394 

945531 

112 

727501 

507 

272499 

54 

7 

673268 

394 

9454154 

113 

727805 

506 

272195 

53 

8 

673505 

394 

945390 

113 

728109 

506 

271891 

52 

9 

673741 

393 

945328 

113 

728412 

506 

27  J  588 

51 

10 

673977 

3U3 

945261 

113 

728716 

506 

271284 

50 

11 

9.674213 

393 

9-945193 

113 

9.729020 

50G 

10.3705)80 

49 

12 

674448 

392 

9451  -25 

113 

729323 

505 

270677 

48  |. 

13 

674684 

392 

945058 

113 

729(526 

505 

270374 

47 

14 

674919 

392 

944990 

113 

729929 

505 

270071 

4(5 

15 

675155 

392 

9449-22 

113 

730333 

505 

859767 

45 

16 

675390 

391 

944854 

113 

730535 

505 

2694(55 

44 

17 

675624 

391 

944786 

113 

730838 

504 

^69162 

43 

18 

675859 

391 

944718 

113 

731141 

504 

268859 

42 

19 

676094 

391 

944650 

113 

731444 

504 

2(58556 

41 

20 

676328 

390 

.  944582 

114 

731746 

504 

268254 

40 

21 

9.676562 

390 

9.944514 

114 

9.732048 

504 

10.267952 

39 

22 

676796 

390 

94444(i 

114 

732351 

503 

267049 

38 

23 

677030 

390 

944377 

114 

732653 

503 

267347 

37 

24 

677264 

389 

944309 

114 

732955 

503 

267045 

36 

25 

677498 

389 

944241 

114 

733257 

503 

266743 

35 

26 

677731 

389 

944172 

114 

733558 

503 

266442 

34 

27 

677964 

388 

944104 

114 

733800 

502 

206140 

33 

28 

678197 

388 

944036 

114 

734162 

502 

205838 

32 

29 

678430 

388 

943967 

114 

734463 

502 

265537 

31 

30 

678663 

388 

943899 

114 

7347(54 

502 

265236 

30 

31 

9.678895 

387 

9.943830 

114 

9.735066 

502 

10.2/54934 

29 

32 

679128 

387 

943761 

114 

735367 

502 

£84633 

28 

33 

679360 

387 

943<>93 

115 

735(568 

501 

264332 

27 

34 

679592 

387 

943024 

115 

735980 

501 

204031 

26 

35 

679824 

386 

943555 

115 

7302(79 

501 

203731 

25 

36 

680056 

386 

94348G 

115 

736570 

501 

263430 

24 

37 

680288 

386 

943417 

115 

730871 

501 

263129 

23 

38 

6805J9 

385 

943348 

115 

737171 

500 

2G2829 

22 

39 

680750 

385 

943279 

115 

737471 

500 

262529 

21 

40 

680982 

385 

943210 

115 

737771 

500 

2G2229 

20 

41 

9.681213 

385 

9.943141 

115 

9.738071 

500 

10.261929 

19 

42 

681443 

384 

943072 

115 

738371 

500 

261029 

18 

43 

681674 

384 

943003 

115 

738071 

499 

261329 

17 

44 

681905 

384 

942934 

115 

738971 

499 

2GI029 

16 

45 

682135 

384 

942864 

115 

739271 

499 

260729 

15 

46 

682365 

383 

942795 

116 

739570 

499 

200430 

14 

47 

682595 

383 

942726 

116 

739870 

499 

2GOI30 

13 

48 

682825 

383 

942056 

116 

7401  09 

499 

259831 

12 

49 

683055 

383 

942587 

116 

740468 

498 

259532 

11 

50 

683284 

382 

942517 

116 

740767 

498 

259233 

10 

51 

9.  683514 

382 

9.942448 

116 

9  741006 

498 

10.258934 

9 

52 

683743 

382 

942378 

116 

741305 

498 

258635 

8 

53 

683972 

382 

942308 

116 

741664 

498 

258336 

7 

54 

684201 

381 

94223!) 

116 

7419(12 

497 

2.18038 

6 

55 

684430 

381 

942109 

11(5 

74-2261 

497 

257739 

5 

56 

684658 

381 

942099 

116 

742559 

497 

257441 

4 

57 

684887 

380 

942029 

116 

742858 

497 

257142 

3 

58 

685115 

380 

941959 

116 

743156 

497 

256844 

2 

59 

685343 

380 

941889 

117 

743454 

497 

256546 

1 

60 

685571 

380 

9-31819 

117 

743752 

496 

256248 

0 

|   Cosine  |      |   Sine       |  Cotang.  |      |   Tang.   M. 

61  Degrees. 


AND  TANGENTS.     (29  Degrees.) 


47 


M. 

Sine     D.     Cosine    D.  |  Tang.     D.     Cotang.  | 

0 

9.685571 

:(-.) 

9.941819 

117 

•J.  743752 

496 

10.256248 

60 

885799 

379 

941749 

117 

744950 

406 

255950 

59 

0 

686037 

379 

941679 

117 

741318 

49'5 

35565S 

58 

3 

689254 

379 

941609 

117 

744(545 

49(5 

255355 

57 

4 

686482 

St9 

941539 

117 

744943 

496 

255057 

56 

5 

IN  57  M 

378 

941469 

117 

745340 

49i> 

854760 

55 

0 

686936 

878 

941398 

117 

745.138 

495 

254462 

54 

7 

687163 

378 

941328 

117 

745835 

495 

254165 

53 

8 

687389 

378 

941258 

117 

74IH3-2 

495 

253868 

52 

9 

687016 

377 

941187 

117 

74ii4-2i) 

495 

253571 

51 

10 

377 

941117 

117 

74;572(j 

495 

253274 

50 

11 

9.686069 

377 

9.941046 

118 

9.747023 

494 

10.252377 

49 

12 

688295 

377 

1)10.17.-) 

118 

747319 

494 

252(581 

48 

13 

683521 

376 

040J05 

118 

747(516 

494 

252384 

47 

14 

688747 

373 

940834 

118 

747913 

494 

252037 

46 

15 

37(5 

9407(53 

118 

748209 

494 

251791 

45 

1(5 

68.)  198 

376 

940693 

118 

748505 

493 

251495 

44 

17 

689423 

375 

940622 

118 

748301 

493 

251199 

43 

18 

639648 

375 

94U551 

118 

749097 

493 

250903 

42 

19 

689873 

375 

940480 

118 

749393 

493 

250607 

41 

20 

69U098 

375 

940409 

118 

749689 

493 

250311 

40 

21 

9.690323 

374 

9.940338 

118 

9.749985 

493 

10.250015 

39 

22 

69J548 

374 

94()267 

118 

750281 

492 

249719 

38 

23 

690772 

374 

940196 

118 

750576 

492 

249424 

37 

24 

69l)i«6 

374 

940125 

119 

750872 

492 

249128 

36 

25 

691230 

373 

94W54 

119 

751167 

492 

248833 

35 

20 

691444 

373 

939932 

119 

7514!52 

492 

248538 

34 

27 

691(568 

373 

939911 

119 

751757 

492 

248243 

33 

28 

691892 

373 

939840 

119 

752052 

491 

247948 

32 

29 

692115 

372 

939768 

119 

752347 

491 

247(553 

31 

30 

093339 

372 

939697 

119 

752(542 

491 

247358   30 

31 

b  6^62 

372 

9.93!K52.-> 

119 

9.732937 

491 

10.247063 

29 

32 

iKKS 

371  - 

939554 

119 

753231 

491 

246769 

28 

33 

.IbM*  {  171 

939482 

119 

7535-26 

491 

246474 

27 

34 

6^:3;  |  VI 

939410 

119 

753820 

490 

246180 

26 

35 

C%>153   -V.l 

939339 

119 

754115 

490 

245885 

25 

36 

."593S57I5  l  'aro 

93!l2ii7 

120 

754409 

490 

245591 

24 

37 

693898  I  i>70 

939195 

'120 

754703 

490 

245297 

23 

38 

694130 

£T0 

9391-2:1 

1-20 

754997 

490 

245003 

22 

39 

594342 

?70 

939052 

120 

755291 

49'J 

244709 

21 

40 

694564 

360     938J8J 

120 

755585 

489 

244415 

20 

41 

C.  694785 

909 

9.938908 

120 

9.755878 

489 

10.244122 

19 

42 

695007 

369 

938336 

120 

756173 

489 

243828 

18 

43 

695229 

3(59 

933763 

120 

751)4(55 

489 

243535 

17 

44 

695450 

368 

938(591 

120 

75(5759 

489 

243241 

16 

45 

695671 

368 

.138619 

120 

757052 

489 

243948 

15 

46 

695892 

368 

638547 

120 

757345 

488 

242(555 

14 

47 

696113 

368 

638475 

120 

757638 

488 

242362 

13 

48 

696334 

367 

933402 

121 

757931 

488 

242069 

12 

49 

696554 

367 

938330 

121 

758224 

488 

241776 

11 

50 

69(3775 

367 

938258 

121 

758517 

488 

241483 

10 

51 

f.  686995 

367 

9.938185 

121 

g.  758810 

488 

10.241190 

9 

52 

697215 

366 

938113 

121 

759102 

487 

240898 

8 

53 

697435 

366 

938040 

I-JI 

759395 

487 

240605 

7 

54 

697654 

366 

937967 

121 

759687 

487 

240313 

6 

55 

697874 

366 

93789S 

121 

759979 

487 

240021 

5 

56 

698094 

365 

937822 

121 

760272 

487 

239728 

4 

57 

898313 

365 

937749 

121 

760564 

487 

239436 

3  I 

58 

698532 

365 

93767(5 

121 

760856 

486 

239144 

2 

59 

698751 

365 

937604 

121 

761148 

486 

238352 

1 

60 

698970 

364 

937531 

121 

761439 

486 

238561 

0 

Cosine  |      |   Sine       |  Cotang.  |      |   Tang.  |  M. 

60  Degress. 


48 


(30  Degrees.)     A  TABLE  OF  LOGARITHMIC 


M.  1   Sine     D   |  Cosine    D.    Tang.   I   D.     Cotang.  | 

0 

9.6118970 

364 

9.937531  1  121 

9.7151439 

486 

10.238561 

60 

1 

699189 

364 

937  4.  58   122 

761731 

486 

59 

2 

6<)9407 

364 

937385 

122 

762033 

486 

237977 

58 

.'{ 

699626 

364 

937312 

122 

762314 

486 

237686 

57 

4 

699844 

363 

937238 

122 

762606 

485 

237394 

56 

5 

700062 

363 

937165 

122 

762897 

485 

237103 

55 

G 

700280 

363 

937092 

122 

763188 

485 

236812 

54 

•; 

700498 

363 

937019 

122 

763479 

485 

231)5-21 

53 

e 

700716 

363 

936946 

122 

703770 

485  • 

236230 

52 

9 

700933 

362 

936872 

122 

764061 

485 

235939 

51 

10 

701151 

362 

936799 

122 

704352 

484 

235648 

50 

u 

9.701368 

362 

9.936725 

122 

9.764643 

484 

10.235:?.57 

49 

12 

701585 

362 

936652 

123 

7(54933 

484 

SfcVMiT   48 

13 

701802 

361 

936578 

123 

765224 

484 

234776   47  '• 

14 

702019 

361 

936505 

123 

765514 

484 

-234481) 

46 

15 

70-2-236 

361 

936431 

123 

765805 

484 

234  J  95 

45 

16 

70-2452 

361 

936357 

123 

766095 

484 

233905 

44 

17 

70-2009 

360 

936284 

123 

766385 

483 

233615 

43 

18 

702885 

360 

936210 

123 

766675 

483 

233325 

42 

19 

703101 

360 

936136 

1-23 

766965 

483 

233035 

41 

20 

703317 

360 

936062 

1-23 

767255 

483 

232745 

40 

21 

9.703533 

359 

9.935988 

123 

9.767545 

383 

10.232455 

39 

'i  22 

703749 

359 

9359M 

123 

767834 

483 

232166 

38 

23 

703964 

359 

935840 

123 

768124 

482 

231876 

37 

24 

704179 

359 

935766 

124 

768413 

482 

231587 

36 

25 

704395 

359 

935692 

124 

768703 

482 

231-297 

35 

26 

704610 

358 

935618 

1-24 

768992 

482 

231008 

34 

27 

704825 

3.58 

935543 

124 

769281 

482 

230719 

33 

28 

705040 

358 

93.54(59 

124 

769570 

482  :    230430 

32 

29 

705254 

358 

935395 

124 

769860 

481 

230140 

31 

30 

705469 

357 

935320 

124 

770148 

481 

229852 

30 

31 

9.705683 

357 

9.935246 

124 

9.770437 

481 

10.229563 

29 

32 

705898 

357 

935171 

124 

770726 

481 

2-29-274 

28 

33 

706112 

357 

935097 

124 

771015 

481 

228985 

27 

34 

7063-20 

356 

9.'io022 

124 

771303 

481 

228697 

26 

35 

706539 

356 

934948 

124 

771592 

481 

228408 

25 

36 

7UU753 

356 

934873 

124 

771880 

480 

2-28120 

24 

37 

706967 

356 

934798 

125 

772168 

480 

227832 

23 

38 

707180 

355 

9317-23 

1-25 

772457 

480 

227543 

22 

39 

707393 

355 

934649 

1'25 

77-274.5 

480 

227255 

21 

40 

707606 

355 

934574 

125 

773033 

480 

2:26967 

20 

41 

9.707819 

355 

9.934499 

125 

9.773321 

480 

10.226679 

19 

42 

708032 

354 

934424 

125 

773608 

479 

226392 

18 

43 

708245 

354 

934349 

125 

773896 

479 

226104 

17 

44 

708458 

354 

934274 

125 

774184 

479 

225816 

16 

45 

708670 

354 

934199- 

125 

774471 

479 

225529 

15 

46 

708882 

353 

934123 

125 

774759 

479 

225241 

14 

47 

709094 

353 

93-1048 

125 

775046 

479 

22^954 

13 

48 

709306 

353 

933973 

125 

77.5:5:j:{ 

479 

224667 

12 

49 

709518 

3.53 

933898 

126 

77.5!  1-21 

478 

224379 

11 

50 

709730 

353 

933822 

126 

775908 

478 

2-24092 

10 

51 

9.70-J941 

352 

9.933747 

126 

9.776195 

478 

10.223805 

9 

52 

710153 

352 

933671 

126 

776482 

478 

223518 

8 

53 

710364 

3.5-2 

933596 

126 

776769 

478 

223231 

7 

54 

710575 

352 

933520 

126 

777055 

478 

2-22945 

6 

55 

710786 

351 

933445 

126 

777342 

478 

222658 

5 

56 

710997 

351 

933309 

126 

777628 

477 

222372 

4 

57 

711208 

351 

933293 

126 

777915 

477 

222085 

3 

58 

711419 

351 

933217 

126 

778201 

477 

221799 

2 

59 

7116-29 

350 

933141 

778487 

477 

221512 

1 

60 

711839 

350 

9330!>6   126    778774 

477 

221226 

0 

|   Cosine  |      |   Sine         Cotang.       |   Tang.  |  M. 

69  Degrees. 


SINKS  AND  TANGENTS.     (31  Degrees.) 


M.  |   Sine     D.   |  Cosine    D.    Tang.   |   D.    Cotang.  | 

0 

9.711839 

350 

«J.i»330!i«5 

126 

9.778774 

477 

10.221236 

60 

1 

713050 

390 

9329:M) 

127 

779060 

477 

220940 

59 

2 

713360 

390 

93-2914 

1-27 

771)316 

476 

220654 

58 

3 

71-246!) 

349 

932838 

127 

779633 

470 

220368 

57 

4 

712(579 

319 

932762 

127 

779918 

476 

220082 

56 

5 

713B89 

349 

932685 

1-37 

780203 

476 

219797 

55 

6 

713098 

349 

932609 

1-27 

780489 

47(5 

219511 

54 

7 

713308 

349 

932533 

127 

781)775 

476 

211)2-25 

53 

8 

718517 

348 

933457 

127 

781060 

476 

218940 

52 

9 

713728 

348 

932380 

1-27 

781346 

475 

218654 

51 

1    ]!) 

713935 

348 

932304 

1-27 

781631 

475 

218369 

50 

II 

9.714144 

348 

9.932228 

127 

9.781916 

475 

10.218084 

49 

13 

714352 

347 

932151 

1*27 

78-2201 

475 

217799 

48 

13 

714561 

347 

932075 

128 

782486 

475 

217514 

47 

14 

7147(59 

347 

931998 

128 

782771 

475 

217229 

46 

15 

714978 

347 

931921 

128 

783056 

475 

216944 

45 

1(5 

71518(5 

347 

931845 

128 

783341 

475 

216659 

44 

«  17 

7  J  5394 

346 

931768 

128 

783626 

474 

918374 

43 

18 

7l.vi()-2 

346 

931691 

128 

783910 

474 

216090 

42 

19 

715809 

346 

931614 

128 

784195 

474 

215805 

41 

20 

710017 

346 

931537 

128 

784479 

474 

215521 

40 

21 

9.716224 

345 

9.931460 

128 

9.784764 

474 

10.215236 

39 

22 

71(143-2 

345 

931383 

128 

785048 

474 

214952 

38 

33 

716(539 

345 

931306 

128 

785332 

473 

214(568 

37 

24 

716846 

345 

931-2-29 

129 

785616 

473 

214384 

36 

25 

717053 

345 

931152 

129 

785900 

473 

214100 

35 

26 

717259 

344 

931075 

129 

786184 

473 

213816 

34 

27 

71746G 

344 

930998 

129 

786468 

473 

213532 

33 

S8 

717673 

344 

930921 

129 

786752 

473 

213248 

32 

29 

717879 

344 

930843 

129 

78703S5 

473 

212964 

31 

30 

718085 

343 

930766 

129 

787319 

472 

212681 

30 

31 

9.718291 

343 

9.930688 

129 

9.787603 

472 

10.212397 

29 

32 

718497 

343 

930611 

129 

787886 

472 

212114 

28 

33 

718703 

343 

930533 

129 

788170 

472 

211830 

27 

34 

718909 

343 

930456 

129 

788453 

472 

211547 

26 

35 

719114 

342 

930378 

129 

783736 

472 

211264   25  ' 

36 

719320 

342 

930301) 

130 

789019 

472 

210981 

24  | 

37 

7195-25 

349 

930323 

130 

789302 

471 

210698 

23 

38 

719730 

342 

930145 

130 

789585 

471 

210415 

22 

39 

7199:55 

341 

930067 

130 

789868 

471 

210132 

21 

40 

T20140 

341 

929989 

130 

790151 

471 

209849 

20 

41 

1).  7-2034.1 

341 

9.929911 

130 

9.790433 

471 

10.209567 

19 

42 

720549 

341 

929833 

130 

790716 

471 

209284 

18 

43 

720754 

340 

929755 

130 

790999 

471 

209001 

17 

44 

7201)58 

340 

929577 

130 

791281 

471 

208719 

16 

45 

7211(52 

340 

929599 

130 

791563 

470 

208437 

15 

46 

7213(56 

340 

929531 

130 

791846 

470 

308154 

14 

47 

7-21570 

340 

!)-2<)44-2 

130 

792128 

470 

307872 

13 

48 

721774 

339 

929364 

131 

792410 

470 

207590 

12 

49 

721978 

339 

929286 

131 

792692 

470 

307308 

11 

50 

722181 

339 

9-29207 

131 

792974 

470 

307026 

10 

51 

9.722385 

339 

9.929129 

131 

9.793-236 

470 

10.206744 

9 

52 

7-2-2.-HH 

339 

929050 

131 

793538 

469 

206462 

8 

53 

722791 

338 

928972 

131 

793819 

469 

306181 

7 

54 

722994 

338 

928893 

131 

794101 

469 

305899 

6 

55 

723197 

339 

928815 

131 

794333 

4(51) 

905617 

5 

56 

733400 

338 

9-28736 

131 

794664 

469 

305336 

4 

57 

337 

938657 

13] 

794915 

469 

905055 

3 

58 

733803 

337 

928578 

131 

795227 

469 

204773 

2 

59 

724007 

337 

9-2849J 

i:u 

795508 

468 

20441)2 

1 

60 

724210 

337 

9-2*120 

131 

795789 

468 

201-211 

0 

|   Cosine          Sine  |     |  Cotang.  |         Tang.  |  M. 

68  Degrees. 

3 

GO 


(32  Degrees.)     A  TABLK  OF  LOGARITHMIC 


M.    Sine   |   D    Cosine   j  D.  |  Tang,   j   D. 

Cotang.  | 

0 

9.724210 

337 

9.938420 

132 

9.7957S9 

468 

10.204-211 

60 

1 

724412 

337 

928342 

132 

796070 

408 

203930 

59 

•2 

724614 

336 

928203 

132 

796351 

468 

203649 

58 

3 

724816 

838 

928183 

132 

79(5(532 

408 

2033li8 

57 

4 

725017 

336 

928104 

132 

796913 

468 

203087 

56 

5 

725219 

336 

928025 

132 

797194 

468 

903806 

55 

-  6 

725420 

335 

92794(5 

132 

797475 

468 

202525 

54 

7 

725622 

335 

9278(57 

132 

797755 

468 

202245 

53 

8 

725823 

335 

927787 

132 

798036 

467 

2019(54 

52 

9 

726024 

335 

927708 

132 

798316 

467 

201684 

51 

10 

726225 

335 

927629 

132 

798596 

467 

201404 

50 

11 

9.726426 

334 

9.927549 

132 

9.798877 

467 

10.201123 

49 

12 

726626 

334 

927470 

133 

799157 

467 

200843 

48 

13 

726H27 

334 

927390 

133 

799437 

407 

200563 

47 

14 

727027 

334 

927310 

133 

799717 

467 

200283 

46 

15 

727228 

334 

927231 

133 

799997 

466 

200003 

45 

16 

727428 

333 

927151 

133 

800277 

466 

199723 

44 

17 

727628 

333 

927071 

133 

800557 

466 

199443 

43 

18 

727828 

333 

926991 

133 

800836 

466 

199164 

42 

19 

728027 

333 

926911 

133 

801116 

466 

198884 

41 

20 

728227 

333 

926831 

133 

801396 

466 

11)8004 

40 

21 

9.728427 

332 

9.926751 

133 

9.801675 

466 

10.198325 

39 

22 

728626 

332 

926671 

133 

801955 

466 

198045 

38 

23 

728825 

332 

926591 

133 

802234 

465 

197766 

37 

24 

729024 

332 

926511 

134 

802513 

465 

197487 

36 

25 

729223 

331 

92(5431 

134 

802792 

465 

197208 

35 

20 

729422 

331 

926351 

134 

803072 

465 

196928 

34 

27 

729621 

331 

926270 

134 

803351 

465 

196649 

33 

28 

729820 

331 

926190 

134 

803630 

465 

196370 

32 

29 

730018 

330 

926110 

134 

803908 

465 

196092 

31 

30 

730216 

330 

926029 

134 

804187 

465 

195813 

30 

31 

9.730415 

330 

9.925949 

134 

9.804466 

464 

10.195534 

29 

32 

730613 

330 

925868 

134 

804745 

464 

195255 

28 

33 

730811 

330 

925788 

134 

805023 

M64 

194977 

27 

34 

731009 

329 

925707 

134 

805302 

464 

194698 

26 

35 

731206 

329 

925626 

134 

805580 

464 

194420 

25 

36 

731404 

329 

925545 

135 

805859 

464 

194141 

24 

37 

731002 

329 

925465 

135 

806137 

4'64 

193863 

23 

38 

731799 

329 

925384 

135 

806415 

463 

193585 

22 

39 

731996 

328 

925303 

135 

806693 

463 

193307 

21 

40 

732193 

328 

925222 

135 

806971 

463 

193029 

20 

41 

9.732390 

328 

9.925141 

135 

9.807249 

463 

10.192751 

19 

42 

732587 

328 

925060 

135 

807527 

•  463 

192473 

18 

43 

732784 

328 

924979 

135 

807805 

463 

192195 

17 

44 

732980 

327 

924897 

135 

808083 

463 

191917 

16 

45 

733177 

327 

92J816 

135 

808361 

463 

191639 

15 

46 

733373 

327 

924735 

136 

808638 

462 

191362 

14 

47 

733569 

327 

924654 

136 

808916 

462 

191084 

13 

48 

733765 

327 

924572 

136 

809193 

4(52 

190807 

12 

49 

733961 

326 

924491 

136 

809471 

462 

1905-29 

11 

50 

734157 

326 

924409 

136 

809748 

462 

190252 

10 

51 

9.734353 

326 

9.924328 

136 

9.810025 

462 

10.189975 

9 

52 

734549 

326 

924246 

136 

810302 

462 

189698 

8 

53 

734744 

325 

924164 

136 

810580 

462 

189420 

7 

54 

734939 

325 

924083 

136 

810857 

462 

189143 

6 

55 

73.1135 

325 

92400  1 

136 

811134 

4G1 

18886(5 

5 

56 

735330 

325 

923919 

136 

811410 

461 

188590 

4 

57 

7355-25 

325 

923837 

136 

811687 

461 

188313 

3 

58 

735719 

324 

&375S 

137 

811964 

461 

188036 

2 

5!) 

735914 

324 

923673 

137 

812241 

461 

187759 

1 

60 

736109 

324 

923591 

137 

812517 

461 

187483 

0 

|   Cosine          Sine   |      Cotang.       |   Tang.  |  M. 

57  Degrees. 


SINES    AND    TANGENTS.       (33    Degrees.') 


M.  j   Sine     D.    Cosine    D.    Tang.   |   D     Totang 

0 

9.  731  i  109 

3-24 

9.9-235U1 

137 

9.  SI  -2:.  17 

401 

10.  J  874  82 

00 

1 

730303 

3-24 

933509 

137 

812794 

401 

187206 

59 

2 

7:«i4!)S 

3-24 

033437 

137 

813070 

401 

180930 

58 

1  3 

73oo<i-2 

323 

923345 

137 

813347 

4(50 

186(553 

57 

4 

73(W86 

3-23 

933263 

137 

813023 

400 

180377 

56 

5 

737080 

3-23 

923181 

137 

813899 

460 

186101 

55 

6 

737-274 

323 

923098 

137 

814175 

460 

185825 

54 

7 

737407 

323 

923016 

137 

814452 

400 

185548 

53 

8 

737861 

322 

9-2-2933 

137 

814728 

400 

185272 

52 

9 

737855 

322 

U22851 

137 

815004 

460 

184996 

51 

10 

738048 

3-22 

922768 

138 

815279 

460 

184721 

50 

11 

9.738241 

322 

9.922686 

138 

9.815555 

459 

10.18-1445 

49 

12 

738434 

3-2-2 

9S9003 

138 

815831 

459 

184109 

48 

13 

738627 

321 

922520 

138 

816107 

459 

183893 

47 

14 

7388-20 

3-21 

9-2-2-138 

138 

816382 

459 

183018 

40 

15 

739013 

321 

922355 

138 

816658 

459 

183342 

45 

1(5 

739-200 

321 

9-2-2-27-2 

138 

816933 

459 

183007 

44 

17 

739398 

321 

922189 

138 

817209 

459 

182791 

43 

18 

739590 

320 

92210(5 

138 

817484 

459 

182516 

42 

19 

739783 

320 

922023 

138 

817759 

459 

182241 

41 

20 

739975 

320 

9-21940 

138 

818035 

458 

181905 

40 

21 

9-740167 

320 

9.921857 

139 

9.818310 

458 

10.181690 

39 

22 

740359 

320 

921774 

139 

818585 

458 

181415 

38 

23 

740550 

319 

921691 

139 

818860 

458 

181140 

37 

24 

74074-2 

319 

981607 

139 

819135 

458 

180805 

36 

25 

740934 

319 

921524 

139 

819410 

458 

180590 

35 

2(5 

741125 

3J9 

921441 

139 

819(584 

458 

180316 

34 

27 

741316 

319 

921357 

139 

819959 

458 

180041 

33 

28 

741508 

318 

921274 

139 

820234 

458 

17G700 

32 

'29 

74  J  699 

318 

921190 

139 

820508 

457 

179492 

31 

30 

741889 

318 

921107 

139 

820783 

457 

179217 

30 

31 

9.742080 

318 

9.921023 

139 

9.821057 

437 

10.178943 

29 

32 

742271 

318 

920939 

140 

821332 

457 

178(508 

28 

33 

742402 

317 

95*0856 

140 

821000 

457 

178394 

27 

34 

742052 

317 

920772 

140 

821880 

457 

178120 

26 

35 

742842 

317 

920088 

140 

822154 

357 

177846 

25 

36 

743033 

317 

920604 

140 

822429 

457 

177571 

24 

37 

743-2-23 

317 

920520 

140 

822703 

457 

177297 

23 

38 

743413 

316 

920436 

140 

822977 

45*5 

177023 

22 

39 

743002 

316 

990353 

140 

823250 

456 

176750 

21 

40 

743792 

316 

920268 

140 

823524 

456 

176476 

20 

41 

9.743982 

316 

9.920184 

149 

9.823798 

456 

10.176202 

19 

42 

744171 

316 

920099 

140 

824072 

456 

175928 

18 

43 

744301 

315 

9-20015 

140 

824345 

456 

175055 

17 

44 

744550 

315 

919931 

141 

824019 

456 

175381 

16 

45 

744739 

315 

919840 

141 

824893 

456 

175107 

15 

4(5 

744928 

315 

919702 

141 

825166 

456 

174834 

14 

47 

745117 

315 

919077 

141 

825439 

455 

174561 

13 

48 

74530(5 

314 

919593 

141 

825713 

455 

174287 

12 

49 

745494 

314 

919508 

141 

825986 

455 

174014 

11 

50 

745C83 

314 

919424 

141 

826259 

455 

173741 

10 

51 

9.745871 

314 

9.919339 

141 

9.826532 

455 

10.173468 

9 

52 

740059 

314 

919254 

141 

826805 

455 

173195 

8 

53 

740248 

313 

919109 

141 

827078 

455 

172922 

7 

54 

74043(5 

313 

919085 

141 

827351 

455 

172(549 

6 

55 

7401  i24 

313 

919000 

141 

827024 

455 

172376 

5 

50 

740812 

313 

918915 

142 

827897 

454 

172103 

4 

57 

74W.H) 

313 

918830 

142 

828170 

454 

171H30 

3 

58 

74"/]87 

312 

918745 

142 

828442 

454 

171558 

2 

59 

747374 

312 

918059 

142 

828715 

454 

171285 

1 

BO 

747502 

312 

918574 

142 

828987 

454 

171013 

0 

Cosine  |      |   Sine        Cotang.  j      |   Tang. 

M.  1 

66  Degrees. 


5-3 


(34  Degree.?.)     A  TABLE  OP  LOGARITHMIC 


M.    Sine     D.   |  Cosine    D.    Tang.   |   D.    Cotang.  | 

0 

9.747562 

312 

9.918574 

142 

9.8-28987 

454 

10.171013 

60  , 

1 

747749 

312 

918489 

142 

8-2U-260 

454 

170740 

59 

2 

747936 

312 

918404 

142 

829532 

454 

170468 

58 

3 

7481-23 

311 

918318 

14-2 

829305 

454 

170195 

57 

4 

748310 

311 

918233 

142 

830077 

454 

169923 

56 

5 

748497 

311 

918147 

14-2 

83:1349 

453 

169651 

55  ! 

6 

748683 

311 

918062 

142 

830H-21 

453 

169379 

54 

.7 

748870 

311 

917976 

143 

830393 

4.33 

169107 

53 

8 

74905(5 

310 

917891 

143 

831165 

453 

168835 

5-2 

9 

749243 

310 

917805 

143 

831437 

453 

168563 

51 

10 

749429 

310 

917719 

143 

831709 

453 

168291 

50 

11 

9.749015 

310 

9.917634 

143 

9.831981 

453 

10.168019 

49 

1-2 

7498J1 

310 

917548 

143 

8322.13 

453 

K17747 

48 

13 

749337 

309 

917415-2 

143 

8325-25 

453 

167475 

47 

14 

750172 

309 

917376 

143 

832796 

453 

167204 

46 

15 

750358 

309 

917-290 

143 

833063 

452 

1(56932 

45 

16 

750543 

309 

917204 

143 

833339 

452 

16(5661 

44 

7.307-29 

309 

917118 

144 

833611 

452 

166389 

43 

18 

750914 

308 

917032 

144 

833882 

4.32 

166118 

42 

19 

751099 

308 

91G946 

144 

834154 

452 

165846 

41 

20 

751284 

303 

916359 

144 

834425 

452 

165575 

40 

21 

9.751469 

308 

9.916773 

144 

9.834696 

452 

10.165304 

39 

22 

7516.34 

308 

916(587 

144 

834957 

4.32 

165033 

38 

23 

751839 

308 

916600 

144 

835-238 

452 

164762 

37 

24 

752023 

307 

916514 

144 

835509 

452 

164491 

36 

25 

752-208 

307 

916427 

144 

835780 

451 

164220 

35 

26 

752:592 

337 

916341 

144 

836051 

451 

163949 

34 

27 

752576 

307 

916254 

144 

836322 

451 

163678 

33 

28 

752760 

307 

916167 

145 

8315593 

4.31 

163407 

32 

29 

752944 

306 

916081 

145 

830864 

451 

163136 

31 

30 

753128 

306 

915994 

145 

837134 

451 

1628t56 

30 

31 

9.753312 

306 

9.915907 

145 

9.8:57405 

451 

10.162595 

29 

32 

753495 

306 

915320 

115 

837675 

451 

182325 

28 

33 

753679 

306 

915733 

145 

837946 

451 

162054 

27 

34 

75386-2 

sas 

915646 

145 

8:58216 

451 

161784 

26 

35 

754046 

305 

915559 

145 

838487 

450 

161513 

25 

23 

7542251 

305 

915472 

145 

838757, 

450 

161243 

24 

37 

754412 

305 

915385 

145 

839027 

450 

160973 

23 

38 

754595 

305 

915-297 

145 

839297 

450 

160703 

22 

39 

754778 

304 

915210 

145 

839568 

450 

160432 

21 

40 

754960 

304 

915123 

146 

839833 

450 

160162 

20 

41 

9.755143 

304 

9.915035 

146 

9.840108 

450 

10.159892 

19 

42 

755326 

304 

914948 

146 

840378 

450 

159622 

18 

43 

755503 

304 

914860 

146 

840647 

450 

159353 

17 

44 

755090 

304 

914773 

146 

840917 

449 

159083 

16 

45 

755872 

303 

914685 

146 

841187 

449 

158813 

15 

46 

7515054 

303 

914598 

146 

841457 

449 

158543 

14 

47 

1*56236 

303 

914510 

146 

841726 

449 

158274 

13 

48 

756418 

303 

914422 

146 

841996 

449 

158004 

12 

49 

756600 

303 

914334 

146 

84-2266 

449 

157734 

11 

50 

756782 

302 

914216 

147 

84-2535 

449 

157465 

10 

51 

9.756963 

302 

9.914158 

147 

9.842805 

449 

10.157195 

9 

52 

757144 

302 

914070 

147 

843074 

419 

15H9-26 

8 

53 

7573-215 

304 

913932 

147 

843343 

449 

156657 

7 

54 

757507 

302 

913334 

147 

843612 

449 

151J388 

6 

55 

757688 

301 

9138;)6 

147 

843882 

448 

156118 

5 

56 

7573S59 

301 

913718 

147 

844151 

448 

155819 

4 

57 

75805!) 

301 

913630 

147 

844420 

448 

155580 

3 

53 

758230 

301 

913541 

147 

844689 

448 

155311 

2 

59 

758411 

301 

913453 

147 

844SI.38 

448 

155042 

1 

60 

758591 

301 

913305 

147 

845227 

448 

154773 

0 

Cosine       |   Sine  |      Cotang.        |   Tang.  |  M. 

55  Degrees. 


SINE-S    AND    TANGKMTS.       (35  Decrees.) 


53 


M.    Sine   |   D.   |  Cosine  |  D.    Tang.     D.    Cotang.  | 

0 

9.758591 

:!oi 

9.9133(55 

147 

9.845-2-27 

448 

10.154773 

60 

1 

758772 

300 

9n-276 

147 

845496 

448 

154504 

59 

2 

758952 

300 

913187 

148 

845764 

448 

154-236 

58 

3 

7r>!H:?-2 

300 

913099 

148 

8415033 

448 

U3987 

57 

4 

759312 

300 

913010 

148 

846302 

448 

153698 

56 

5 

7.-i!l  Kt-2 

300 

913992 

148 

84(5570 

447 

153430 

55  1 

6 

759672 

299 

912833 

148 

846839 

447 

153161 

54 

7 

759852 

299 

91-2744 

118 

847107 

447 

152893 

53 

8 

7601)31 

2i)J 

91-26.1,-) 

148 

847376 

447 

152824 

52 

B 

768211 

299 

9125(56 

148 

817*544 

447 

15-2356 

51 

10 

760391) 

289 

912477 

148 

847913 

447 

152087 

50 

11 

9.760569 

298 

9.912388 

148 

9.848181 

447 

10.151819 

49 

12 

760748 

298 

91  2-299 

149 

848449 

447 

151551 

48 

13 

760927 

298 

91  -2-2  10 

149 

848717 

447 

151283 

47 

14 

761106 

298 

912121 

149 

848986 

447 

151014 

46 

15 

7(51285 

298 

912031 

149 

849254 

447 

150746 

45 

16 

761464 

298 

911942 

149 

84D.V2-2 

447 

150478 

44 

17 

76164-2 

297 

911853 

149 

849790 

446 

150210 

43 

18 

761821 

297 

911763 

149 

850058 

446 

149942 

42 

19 

761999 

297 

911674 

149 

8.VW25 

446 

149075 

41 

20 

762177 

297 

911584 

149 

850593 

446 

149407 

40 

21 

9.762356 

297 

9.911495 

149 

9.850881 

446 

10.149139 

39 

22 

762534 

296 

91140.') 

149 

851129 

446 

148871 

38 

23 

7(5-27  1-2 

296 

911315 

150 

851396 

446 

148604 

37 

24 

762889 

MB 

9112-26 

150 

851664 

446 

148336 

36 

25 

763067 

296 

911136 

150 

851931 

446 

J  430(59 

35 

26 

7f:3-245 

296 

911046 

150 

852199 

446 

147801 

34 

27 

715:142-2 

296 

<)10<).->6 

150 

85-21(56 

446 

147534 

33 

28 

763r>i>n 

295 

910866 

150 

8.12733 

445 

1472(57 

32 

29 

7153777 

295 

910776 

150 

853001 

445 

146999 

31 

30 

763954 

295 

910686 

150 

8532G8 

445 

146732 

30 

31 

9.764131 

295 

9.910596 

150 

9.853535 

445 

10.146465 

29 

32 

764308 

295 

910506 

150 

853802 

445 

146198 

28 

33 

764485 

294 

910415 

150 

854069 

445 

145931 

27 

34 

7646(52 

294 

9103-25 

151 

&J4336 

445 

145664 

26 

35 

764838 

294 

910235 

151 

854'5U3 

445 

145397 

25 

30 

7fi5:)15 

2i)4 

910144 

151 

854870 

445 

145130 

24 

37 

7(55191 

294 

910054 

151 

855137 

445 

144863 

23 

38 

765367 

294 

<);><>963 

151 

855404 

445 

144596 

22 

39 

765544 

293 

909873 

151 

855671 

444 

144329 

21 

40 

765720 

293 

909782 

151 

855938 

444 

144063 

20 

41 

9.765896 

293 

9.90%91 

151 

9.856-204 

444 

10.1437% 

19 

42 

766072 

293 

909601 

151 

856471 

444 

143.V29 

18 

43 

71H5247 

293 

909510 

151 

856737 

444 

143-2(53 

17 

44 

7(56423 

293 

909419 

151 

857004 

444 

142996 

16 

45 

765598 

292 

909328 

152 

H57270 

444 

142730 

15 

46 

766774 

292 

903237 

152 

H.-)7.-)37 

444 

142463 

14 

47 

786949 

29-2 

909146 

152 

857803 

444 

14-2197 

13 

40 

767124 

292 

90.4055 

152 

858069 

444 

141931 

12 

49 

767300 

292 

903964 

152 

858336 

444 

141664 

11 

50 

767475 

291 

908873 

152 

858602 

443 

141398 

10 

51 

^.  767649 

291 

9.908781 

152 

9.858868 

443 

10.141132 

9 

52 

767824 

291 

9^81590 

152 

859134 

443 

140866 

8 

53 

767999 

291 

908599 

152 

859400 

443 

14!)600 

7 

54 

768173 

291 

909507 

IM 

8596(56 

443 

140334 

6 

55 

768348 

290 

908416 

153 

859932 

443 

140068 

5 

56 

768522 

290 

908334 

L53 

860153 

443 

4 

57 

768697 

290 

90H-233 

153 

800464 

413 

139536 

3 

58 

768371 

290 

908141 

153 

8-10730 

443 

139-270 

2 

59 

769043 

290 

908049 

153 

8f>0995 

443 

i3!>:)ti.-> 

1 

60 

769219 

290 

907958 

153 

861261 

443 

138739 

0 

1    |  Cosine       |   Sine  |    |  Cotang.  |      |   Tang.  |  M. 

54  Degrees. 


54 


(3G  Degrees.)     A  TABLE  or  LOGARITHMIC 


M.    Sine   |   D     Cosine   |  D.    Tang.     D.   |   Cotang.      i 

0 

9.769219 

290 

9.907953 

153 

9.861-261 

443 

10.138739 

60 

1 

769393 

289 

9J7866 

153 

861527 

443 

138473 

59 

o 

709566 

289 

907774 

153 

8(51792 

442 

138-208 

58 

3 

769740 

289 

907682 

153 

803058 

442 

137942 

57 

4 

769913 

289 

907590 

153 

862323 

442 

137677 

56 

5 

770087 

289 

907498 

153 

8(5-2589 

442 

137411 

55 

6 

770260 

288 

9:»7406 

153 

8(52854 

442 

137146 

54 

7 

770433 

288 

907314 

154 

8(53119 

442 

130881 

53 

8 

770606 

288 

907222 

154 

863385 

442 

130615 

52 

9 

770779 

288 

907129 

154 

863f»50 

442 

130350 

51 

10 

770952 

288 

907037 

154 

863915 

442 

136085 

50 

11 

9.771125 

288 

9.906945 

154 

9.864180 

442 

10.135820 

49 

12 

771298 

287 

906852 

154 

864445 

442 

135555 

48 

13 

771470 

287 

906760 

154 

8(54710 

442 

135-290 

47 

14 

771643 

287 

906667 

154 

864975 

441 

135025 

4(5 

15 

771815 

287 

906575 

154 

865240 

441 

134700 

45 

16 

771987 

287 

906482 

154 

865505 

441 

134495 

44 

17 

772159 

287 

906389 

'  155 

8(55770 

441 

134230 

43 

18 

772331 

286 

906296 

155 

8(56035 

441 

1339(55 

42 

19 

772503 

288 

906204 

155 

866300 

441 

133700 

41 

20 

772675 

286 

906111 

155 

866564 

441 

133436 

40 

21 

9.772847 

286 

9.90C018 

155 

9.8668-29 

441 

10.133171 

39 

22 

7730  J  8 

286 

905925 

155 

867094 

441 

132906 

38 

|  23 

773190 

286 

905832 

155 

867358 

441 

132642 

37 

24 

773361 

285 

905739 

155 

867623 

441 

132377 

36 

25 

773533 

285 

905645 

155 

867887 

441 

132113 

35 

26 

773704 

285 

905552 

155 

868152 

440 

131848 

34 

27 

773875 

285 

905459 

155 

868416 

440 

131584 

33 

28 

774046 

285 

905366 

156 

868680 

440 

131320 

32 

29 

774217 

2H5 

905272 

156 

868945 

440 

131055 

31 

30 

774388 

284 

905179 

156 

869209 

440 

130791 

30 

31 

9.774558 

284 

9.905085 

156 

9.869473 

440 

10.130527 

23 

32 

774729 

284 

904992 

156 

869737 

440 

130-263 

28 

33 

77-1899 

284 

904898 

156 

870001 

440 

129999 

27 

34 

775070 

284 

904804 

156 

870-265 

440 

129735 

26 

35 

775240 

284 

9047]  1 

156 

870529 

440 

129471 

25 

36 

775410 

283 

904617 

156 

870793 

440 

129-207 

24 

37 

'  775580 

283 

904523 

156 

871057 

440 

128943 

23 

36 

775750 

283 

904429 

157 

871321 

440 

128679 

22 

39 

775920 

283 

904335 

157 

871585 

440 

128415 

21 

40 

77609!) 

283 

904241 

157 

871849 

439 

128151 

20 

41 

9.776259 

283 

9.904147 

157 

9.872112 

439 

10.127888 

19 

42 

776429 

282 

9JMO.-.3 

157 

872376 

439 

127624 

18 

43 

776598 

282 

903959 

157 

87-2640 

439 

1273(50 

17 

44 

776768 

282 

903864 

157 

872903 

439 

1-27097 

16 

45 

776937 

282 

903770 

157 

873167 

439 

126833 

15 

46 

777106 

282 

903676 

157 

873430 

439 

1  20570 

14 

47 

777275 

281 

903581 

157 

873694 

439 

126306 

13 

48 

777444 

281 

903487 

157 

873957 

439 

120043 

12 

49 

777613 

281 

903392 

158 

874220 

439 

125780 

11 

50 

777781 

281 

903298 

158 

874484 

439 

125516 

10 

51 

9.777950 

281 

9.903203 

158 

9.874747 

439 

10.125253 

9 

52 

778119 

281 

903108 

158 

875010 

439 

124990 

8 

53 

778287 

280 

903014 

158 

875273 

438 

124727 

7 

54 

778455 

280 

902919 

158 

875536 

438 

124464 

6 

55 

778624 

280 

902824 

158 

875800 

438 

1-24200 

5 

56 

778792 

280 

902729 

158 

876003 

438 

123937 

4 

57 

778900 

280 

90-2634 

158 

876326 

438 

1-23674 

3 

58 

779128 

280 

902539 

159 

876589 

438 

l-2:!4  11 

2 

59 

779295 

279 

91)2444 

159 

876851 

438 

123149 

1 

CO 

779463 

279 

902349 

159 

877114 

438 

12-2886 

0 

|   Cosine  |      |   Sine   |     |  Cotang.  |      |   Tang.   M.  1 

53  Degrees. 


SINES  AND  TANGENTS.     (37  Degrees.) 


55 


M.  |   Sine     D.    Cosine    D.    Tang.     D     Cotang 

0 

9.779463 

279 

9.9:  12311) 

I.V.I 

9.B77114 

438 

10.122846 

t50 

J 

779831 

279 

902-253 

159 

877377 

438 

122633 

59 

i 

779798 

379 

90-21.  )8 

156 

877(540 

438 

123360 

58 

3 

7799(36 

279 

90-2063 

159 

877903 

438 

12-2097 

57 

4 

780133 

279 

901%7 

158 

8781(55 

438 

121835 

56 

5 

7f<()3l)0 

278 

901872 

159 

8784-28 

438 

121572 

55 

6 

780467 

278 

901776 

159 

878(591 

438 

121309 

54 

7 

7rtli(i34 

278 

DOI6HI 

158 

878953 

437 

121047 

53 

8 

780801 

278 

901535 

159 

879216 

437 

120784 

52 

9 

780J68 

278 

91)1490 

159 

879478 

437 

120522 

51 

10 

781134 

278 

'JO  1394 

160 

879741 

437 

120259 

50 

11 

9.781301 

277 

9.901293 

160 

9.83,)003 

437 

10.119937 

49 

13 

7H14t« 

277 

901202 

160 

88,)2iij 

437 

119735 

48 

13 

781634 

277 

901105 

160 

880528 

437 

119472 

47 

14 

781800 

277 

901010 

160 

880790 

437 

119210 

46 

15 

781966 

277 

900914 

160 

881052 

437 

118948 

45 

16 

78-2132 

277 

900S18 

160 

881314 

437 

118686 

44 

17 

782-298 

276 

900722 

160 

881576 

437 

118424 

43 

18 

782464 

276 

9i)0i»26 

160 

881839 

437 

118161 

42 

19 

782830 

276 

900529 

160 

882101 

437 

117899 

41 

20 

782796 

276 

901)433 

161 

88-3363 

436 

117637 

40 

21 

9.782961 

276 

9.900337 

161 

9.882625 

436 

10.117375 

39 

22 

783127 

276 

900240 

161 

882887 

436 

117113 

38 

23 

783292 

275 

900144 

161 

883148 

436 

116852 

37 

24 

783458 

275 

900047 

161 

883410 

436 

116590 

36 

25 

783623 

275 

899951 

161 

883672 

436 

116328 

35 

26 

783788 

275 

893854 

161 

8839:14 

436 

116066 

34 

27 

783953 

275 

899757 

161 

884196 

436 

115804 

33 

28 

784118 

275 

8991)60 

161 

884457 

436 

115543 

32 

29 

784282 

274 

899564 

161 

884719 

436 

115281 

31 

30 

784447 

274 

899467 

162 

884980 

436 

115020 

30 

:u 

9.784612 

274 

9.899370 

162 

9.885242 

436 

10.114758 

29 

M 

784776 

274 

899273 

162 

885503 

436 

114497 

28 

33 

784941 

274 

899176 

162 

885765 

436 

114235 

27 

34 

785105 

274 

899078 

162 

88(50-26 

43(5 

113974 

26 

3o 

785369 

273 

898981 

162 

886-288 

436 

113712 

25 

36 

785433 

273 

898884 

162 

881  M  49 

435 

113451 

24 

37 

785507 

273 

898787 

162 

886810 

435 

113190 

23 

38 

785761 

273 

898689 

162 

887072 

435 

112928 

22 

39 

785925 

273 

898592 

162 

887333 

435 

112667 

21 

40 

786089 

273 

898494 

163 

887594 

435 

112406 

20 

41 

9.7H(r252 

272 

9.898397 

163 

9.887855 

435 

10.112145 

19 

42 

786416 

272 

898299 

163 

888116 

435 

111884 

18 

43 

786579 

272 

89820-2 

163 

888377 

435 

111623 

17 

44 

786742 

272 

898104 

163 

888639 

435 

111361 

16 

45 

786906 

272 

898006 

163 

888900 

435 

1]1100 

15 

46 

7871)69 

272 

897908 

163 

8891(50 

435 

110840 

14 

47 

787332 

271 

897810 

163 

889421 

435 

110579 

13 

48 

787395 

271 

897712 

163 

889(582 

435 

110318 

12 

49 

787557 

271 

897614 

163 

889943 

435 

110057 

11 

.50 

787720 

271 

897516 

163 

890204 

434 

109796 

10 

51 

9.787883 

271 

9.897418 

164 

9.890465 

434 

10.109535 

9 

52 

788845 

271 

897320 

164 

890725 

434 

109-275 

8 

53 

788208 

27! 

897222 

164 

890986 

434 

109014 

7 

54 

788370 

270 

897123 

164 

891247 

434 

108753 

6 

55 

783532 

270 

897025 

164 

891507 

434 

108493 

5 

56 

788694 

270 

896926 

164 

8917158 

434 

108232 

4 

57 

788856 

270 

896828 

164 

434 

107972 

3 

58 

7891)18 

270 

896729 

164 

892289 

434 

107711 

2 

59 

789180 

270 

89(5631 

164 

892549 

434 

107451 

1 

60 

7*9342 

26J 

896532 

164 

892810 

434 

107190 

0 

|  Cosine  |         Sine   |    |  Cotang.       |   Tang.    M.  j 

56 


(38  Degrees.)     A  TABLE  OP  LOGARITHMIC 


M.  |   Sine     D   I  Cosine    D,  |  Tang.     D.   |  Cotang.  | 

0 

9.78934-2 

980 

9.891)532 

Ifi4 

9.892810 

434 

10.107190  I  60 

1 

789504 

369 

896433 

165 

893070 

434 

106930 

59 

2 

789665 

269 

896335 

165 

893331 

434 

106669 

58 

3 

7898-27 

269 

896236 

165 

893591 

434 

106409 

57 

4 

789988 

269 

896137 

lt>5 

893851 

434 

106149 

56 

5 

790149 

269 

896038 

165 

894111 

434 

105889 

55 

6 

790310 

268 

895939 

165 

894371 

434 

105629 

54 

7 

7!K)471 

268 

895840 

165 

8:)4i)32 

433 

105368 

53 

8 

790632 

268 

895741 

1G5 

894892 

433 

105108 

52 

9 

790793 

208 

895641 

165 

895152 

433 

104848 

51 

10 

790954 

268 

895542 

165 

895412 

433 

104588 

50 

11 

9.791115 

268 

9.895443 

166 

9.895672 

433 

10.104328 

49 

12 

791275 

267 

895343 

166 

895932 

433 

104068 

48 

13 

791436 

267 

895244 

166 

896192 

433 

103808 

47 

14 

791596 

267 

895145 

166 

8l>6452 

433 

103548 

46 

15 

791757 

267 

895045 

166 

896712 

433 

103288 

45 

16 

791917 

267 

894945 

166 

896971 

433 

103029 

44 

17 

792077 

367 

894846 

166 

897231 

433 

102769 

43 

18 

792237 

266 

894746 

166 

897491 

433 

102509 

42 

19 

792397 

266 

894646 

166 

897751 

433 

10-2249 

41 

20 

792557 

266 

894546 

166 

898010 

133 

101990 

40 

21 

9.792716 

266 

9.894446 

167 

9.?!i,-}?0 

433 

10.101730 

39 

22 

792876 

266 

894346 

167 

898530 

433 

101470 

38 

23 

793035 

2fi6 

894246 

167 

898789 

433 

101211 

37 

24 

793195 

265 

894146 

167 

899049 

432 

100951 

36 

25 

793354 

265 

894046 

167 

899308 

432 

100692 

35 

26 

793514 

265 

893946 

167 

899568 

432 

100432 

34 

27 

793673 

205 

893846 

167 

899827 

432 

100173 

33 

28 

793332 

265 

893745 

167 

900086 

432 

099914 

32 

29 

793991 

265 

893645 

167 

900346 

432 

099654 

31 

30 

794150 

264 

893544 

167 

900605 

432 

099395 

30 

31 

9.794308 

264 

9.893444 

168 

9.900864 

432 

10.099136 

29 

32 

794467 

264 

893343 

168 

901124 

432 

098876 

28 

33 

794626 

2154 

89*243 

1<!8 

901383 

439 

098617 

27 

34 

794784 

2<i4 

893142 

168 

901642 

432 

098358 

2fi 

35 

794942 

264 

89IJ041 

168 

901901 

432 

098099 

25 

36 

795101 

2G4 

892940 

168 

902160 

432 

097840 

24 

37 

795259 

263 

892839 

168 

902419 

432 

097581 

23 

38 

795417 

263 

892739 

168 

900679 

432 

097321 

22 

39 

795575 

263 

892638 

168 

902938 

432 

097062 

21 

40 

795733 

263 

892536 

168 

903197 

431 

096803 

20 

41 

9.795891 

263 

9.892435 

169 

9.903455 

431 

10.09(5545 

19 

42 

796049 

263 

892334 

169 

9H37  14 

431 

096286 

18 

43 

796206 

263 

892233 

169 

903973 

431 

096027 

17 

44 

796364 

262 

892132 

169 

904232 

431 

095768 

16 

45 

796521 

901 

892030 

169 

904491 

431 

095509 

15 

46 

796679 

262 

891929 

169 

904750 

431 

095250 

14 

47 

796836 

262 

891827 

169 

905008 

431 

094992 

13 

48 

79R993 

262 

891726 

169 

9052(57 

431 

094733 

12 

49 

797150 

261 

.  891624 

1(59 

905526 

431 

094474 

11 

50 

797307 

261 

891523 

170 

905784 

431 

094216 

10 

51 

9.797464 

261 

9-891421 

170 

9.906043 

431 

10.093957 

9 

52 

797621 

261 

891319 

170 

906302 

431 

093698 

8 

53 

797777 

261 

891217 

170 

900560 

431 

093440 

7 

54 

797934 

261 

891115 

170 

906819 

431 

093181 

6 

55 

798091 

261 

891013 

170 

907077 

431 

092923 

5 

56 

798247 

261 

890911 

170 

907336 

431 

002664 

4 

57 

798403 

260 

8W889 

170 

907594 

431 

092406 

3 

58 

798560 

2liO 

890707 

170 

907852 

431 

092148 

2 

59 

798716 

260 

890605 

170 

908111 

430 

091889 

1 

60  1   798872 

260 

890503 

170 

908369 

430 

091631 

0 

|   Cosine       |   Sine   |     |  Cotang.  |      |   Tang.  |  M. 

SINRS    AMI    TANGKNTS.       (39    Degrees.) 


5? 


M. 

Sine   |   D.     Cosine  |  D.    Tang.     D.     Cotang  | 

0 

9.798872 

260 

9.890503 

170 

9.908369 

430 

W.i  '.lira  i 

60 

1 

799028 

260 

890400 

171 

908628 

430 

091372 

59 

o 

798184 

2tiO 

890298 

171 

908886 

430 

091114 

58 

3 

799339 

259 

890195 

171 

909144 

430 

090856 

57 

4 

799495 

259 

890093 

171 

909402 

430 

090598 

56 

5 

799851 

299 

889990 

171 

909660 

430 

090340 

55 

6 

799806 

•..'.V.I 

171 

909918 

430 

090082 

54 

7 

799963 

399 

171 

'.110177 

4:w 

089823 

53 

8 

800117 

250 

889882 

171 

910439 

430 

089565 

52 

9 

800272 

258 

889579 

17! 

910693 

430 

089307 

51 

10 

800427 

258 

889477 

171 

910051 

430 

089049 

50 

11 

9.800582 

258 

9.8«9374 

172 

9.911209 

430 

10.088791 

49 

12 

800737 

258 

889271 

172 

911467 

430 

088533 

48 

13 

800899 

258 

889  J  68 

179 

<nn-.M 

430 

088276 

47 

14 

Hill  047 

258 

889064 

172 

911982 

430 

088018 

46 

15 

801301 

258 

888961 

172 

912240 

430 

0877(50 

45 

16 

801356 

257 

886858 

172 

912498 

430 

087502 

44 

17 

801511 

257 

888755 

172 

912756 

430 

087244 

43 

18 

801665 

xT.7 

883*651 

172 

913014 

429 

086986 

42 

19 

801819 

257 

888548 

172 

913271 

429 

086129 

41 

20 

801973 

257 

888444 

173 

913529 

429 

086471 

40 

21 

9.802128 

257 

9-883341 

173 

9.913787 

429 

10.086213 

39 

22 

802282 

256 

888-237 

173 

914044 

429 

085956 

38 

23 

802436 

2.~>6 

888134 

173 

914302 

429 

085698 

37 

24 

802589 

256 

888030 

173 

914560 

429 

085440 

36 

35 

802743 

256 

887926 

173 

914817 

429 

085183 

35 

26 

802897 

256 

887822 

173 

915075 

429 

084925 

34 

27 

803050 

2.Vi  • 

887718 

173 

915332 

429 

OH46«8 

33 

28 

803-204 

256 

887614 

173 

915590 

429 

084410 

32 

29 

803357 

255 

887510 

173 

915847 

429 

084153 

31 

30 

803511 

255 

887406 

174 

916104 

429 

083896 

30 

31 

9-803664 

255 

9.887302 

174 

9.916362 

429 

10.083638 

29 

32 

803817 

255 

887198 

174 

916619 

429 

083381 

28 

33 

803970 

.  255 

887093 

174 

916877 

429 

083123 

27 

34 

8(14123 

255 

886989 

lf4 

917134 

429 

082866 

26 

35 

804276 

254 

886885 

174 

917391 

429 

082609 

25 

36 

80443» 

254 

886780 

174 

ill  7048 

429 

082352 

24 

37 

804581 

254 

886676 

174 

917905 

429 

082005 

23 

38 

804734 

254 

886571 

175 

918163 

428 

081837 

22 

39 

804886 

254 

886466 

174 

918420 

428 

081580 

21 

40 

805039 

254 

886362 

175 

918677 

428 

081323 

20 

41 

9--805191 

254 

9.886257 

175 

9.918934 

428 

10.081066 

19 

42 

805343 

253 

886152 

175 

919191 

428 

080809 

18 

43 

805495 

253 

886047 

175 

919448 

080552 

17 

44 

805647 

253 

885942 

175 

919705 

428 

080295 

16 

45 

805799 

253 

885837 

175 

919962 

428 

080038 

15  1 

46 

805951 

253 

885732 

175 

920219 

428 

079781 

I*  . 

47 

806103 

253 

885627 

175 

920476 

428 

079524 

13 

48 

806254 

253 

885522 

175 

920733 

428 

079267 

12 

49 

806406 

252 

885416 

175 

920990 

428 

079010 

11 

50 

806557 

252 

885311 

176 

921247 

428 

078753 

10 

51 

9-806709 

252 

9.885205 

176 

9.921503 

428 

10.078497 

9 

52 

896800 

252 

885100 

176 

921760 

428 

078240 

8 

53 

807011 

252 

884994 

176 

922017 

428 

077983 

7 

54 

807163 

252 

884889 

176 

922274 

428 

077726 

6 

55 

807314 

252 

884783 

176 

922530 

428 

077470 

5 

56 

807465 

251 

884677 

176 

922787 

428 

077213 

4 

57 

807615 

251 

884572 

176 

9-23044 

428 

076956 

3 

58 

807766 

251 

884466 

176 

923300 

428 

076700 

2 

59 

807917 

251 

884360 

176 

923557 

427 

076443 

1 

60 

808067 

251 

884954 

177 

923813 

427 

076187 

0 

|  Cosine        |   Sine   |      Cotang.          Tang. 

M.  , 

60  Degree*. 


58 


(40  Degrees.)     A  TABF.E  or  LOGARITHMIC 


M.  |   Sine   |   D.   |  Cosine    D.    Tang.   |   D.     Cotang.  | 

0 

9.808067 

251 

9.884254 

177 

9.923813 

4-27 

10.076187 

60 

1 

8)8218 

251 

884148 

177 

9-2-1070 

427 

075930 

59 

2 

8033:  £ 

251 

884042 

177 

9-213-27 

4-27 

075;i73 

58  . 

3 

808519 

250 

a3393!> 

177 

924583 

427 

075417 

57 

4 

8J8I569 

250 

8833-29 

177 

9-2-1810 

427 

075160 

56 

5 

808819 

250 

883723 

177 

<J-25ii96 

4-27 

074904 

55 

6 

808989 

250 

883(517 

177 

9-2535-2 

427 

074648 

54 

809119 

250 

883510 

177 

9-256U9 

427 

0743'Jl 

53 

8 

809-269 

250 

833404 

177 

9258:55 

427 

074135 

52 

9 

809419 

249 

083-2U7 

173 

92151-22 

427 

073878 

51 

10 

809569 

249 

8;31<J1 

178 

'J-2G373 

427 

073622 

50 

11 

9.809718 

249 

9.833084 

178 

9.92G634 

427 

10.07:<366 

49 

12 

809868 

249 

882977 

178 

(J2(>890 

4-27 

073110 

48 

13 

810017 

249 

832371 

178 

027147 

427 

072353 

47 

14 

810167 

249 

832764 

178 

927403 

427 

072.)97 

46 

15 

810316 

248 

883S57 

178 

927659 

427 

072341 

45 

16 

810465 

248 

8d25.50 

178 

9-27915 

.  427 

07-2085 

44 

17 

810614 

248 

882443 

173 

928171 

427 

071829 

43 

18 

810763 

248 

882336 

179 

923427 

4-27 

071573 

42 

19 

810912 

248 

882-2-29 

179 

928683 

427 

071317 

41 

20 

811061 

248 

882121 

179 

928940 

427 

071060 

40 

21 

9.811210 

248 

9.832014 

179 

9.929196 

427 

10.070804 

39 

22 

811358 

247 

881907 

179 

929452 

427 

070548 

38 

23 

811507 

247 

831799 

179 

9-29708 

427 

070292 

37 

24 

811655 

247 

831692 

179 

929964 

4-26 

070036 

36 

,25 

811804 

247 

881584 

179 

930-220 

4-26 

069780 

35 

26 

8J  1952 

247 

88  J  477 

179 

930475 

426 

01595-25 

34 

27 

812100 

247 

831369 

179 

9:50731 

426 

0159-269 

33 

28 

812-248 

247 

881261 

180 

93:)987 

4-26 

069013 

32 

29 

812396 

246 

881153 

180 

931243 

426 

068757 

31 

30 

812544 

246 

881046 

180 

931499 

426 

008501 

30 

31 

9.812692 

246 

9.880938 

180 

9.931755 

426 

10.068245 

29 

32 

81-2840 

246 

880830 

180 

93)010 

426 

067990 

28 

33 

812938 

246 

88072-2 

180 

93-2266 

426 

067734 

27 

34 

813135 

246 

880613 

180 

93-J5-2-2 

426 

067478 

26 

35 

81  3-283 

246 

880505 

180 

93-2778 

426 

067-222 

25 

36 

813430 

245 

880397 

180 

933(133 

426 

.  06(59117 

24 

37 

813578 

245 

880239 

181 

933289 

426 

066711 

23 

38 

813725 

245 

880180 

181 

933545 

426 

066455 

22 

i  :<'.( 

813872 

245 

880072 

181 

933800 

426 

066-200 

21 

40 

814019 

245 

879963 

181 

934056 

426 

065944 

20 

41 

9.814166 

245 

9.879855 

181 

9.934311 

426 

10.065689 

19 

42 

814313 

245 

879746 

181 

934567 

426 

0(55433 

18 

43 

814460 

244 

879637 

181 

934823 

42f) 

065177 

17 

44 

814607 

244 

879529 

181 

935078 

426 

064922 

16 

45 

814753 

iM4 

879  '20 

181 

935333 

426 

0(54(567 

15 

46 

814900 

244 

879311 

181 

935589 

426 

064411 

14 

47 

815046 

244 

879202 

182 

935844 

426 

064156 

13 

48 

815193 

244 

879:  193 

182 

93(5100 

426 

063900 

12 

49 

815339 

244 

878984 

182 

9:56355 

426 

063(145 

11 

50 

815485 

243 

878875 

182 

936610 

426 

063390 

10 

51 

9.815631 

243 

9.878766 

18-2 

9-936866 

425 

10.063134 

9 

52 

815778 

243 

878656 

182 

937121 

425 

062879 

8 

53 

815924 

243 

878547 

182 

937376 

425 

062624 

7 

54 

816069 

243 

878438 

182 

937632 

425 

0623(58 

6 

55 

816215 

243 

878328 

182 

937887 

425 

015-2113 

5 

56 

816361 

243 

878219 

183 

938142 

425 

061858 

4 

57 

816507 

242 

878109 

183 

938398 

425 

061602 

3 

.V 

816652 

242 

877999 

183 

938653 

425 

061347 

2 

59 

816798 

242 

877890 

183 

938908 

425 

001092 

1 

60 

816943 

242 

877780  |  183 

939163 

425 

060837 

0 

Cosine          Sine  |      Cotang.  |      |   Tang.  |  M. 

49  Degrees. 


SINES    AND   TANGENTS.       (41   Degrees.) 


M.  |   Sine   |   D    Cosine    D.  |  Tang.     D.     Cotang.  1 

8 

9.81(5943 

242 

D.H77780 

183 

9.H39U53 

425 

10.060837 

60 

1 

817088 

242 

877670 

183 

i  13!  141* 

425 

060582 

59 

2 

817233 

242 

877/iW) 

183 

93<W73 

425 

060327 

58 

3 

817379 

242 

877450 

183 

WJil-JH 

425 

060072 

57 

4 

817524 

241 

877340 

183 

940183 

425 

059817 

56 

5 

817668 

241 

877230 

184 

940438 

425 

059562 

55 

G 

817813 

241 

877120 

184 

940694 

425 

059306 

54 

7 

817958 

241 

877010 

184 

940949 

425 

059051 

53 

8 

818103 

241 

87fi8!>9 

184 

941204 

425 

058796 

52 

9 

818247 

241 

876789 

184 

941453 

425 

058542 

51 

10 

818392 

241 

876678 

184 

941714 

425 

058286 

50 

11 

9,818536 

240 

9.876568 

184 

9.941968 

425 

10.058032 

49 

12 

818IJ81 

240 

876457 

184 

942223 

425 

057777 

48 

13 

818825 

240 

876347 

184 

942478 

425 

057522 

47 

14 

818969 

240 

876236 

185 

942733 

425 

057267 

46 

15 

819113 

240 

876125 

185 

942988 

425 

057012 

45 

16 

819257 

240 

876014 

185 

943243 

425 

056757 

44 

17 

819401 

240 

875904 

185 

943498 

425 

056502 

43 

18 

819545 

230 

875793 

185 

943752 

425 

056*48 

42 

19 

819689 

239 

875682 

185 

944007 

425 

055993 

41 

20 

819832 

239 

875571 

185 

944262 

425 

055738 

40 

21 

9-819976 

239 

9.875459 

185 

9,944517 

425 

10.055483 

39 

22 

820120 

239 

875348 

185 

944771 

424 

055229 

38 

23 

820263 

239 

875237 

185 

945026 

424 

054974 

37 

24 

820406 

239 

875126 

186 

945281 

424 

054719 

36 

25 

820550 

238 

875014 

186 

945535 

424 

054465 

35 

26 

820693 

238 

874903 

186 

945790 

424 

054210 

34 

27 

820836 

238 

874791 

186 

946045 

424 

053955 

33 

28 

820979 

238 

874680 

186 

946299 

424 

053701 

32 

29 

821122 

238 

874568 

186 

946554 

424 

053446 

31 

30 

821265 

238 

874456 

186 

946808 

424 

053192 

30 

31 

9.821407 

238 

9.874344 

186 

9.947063 

424 

10.052937 

29 

32 

821550 

238 

874232 

187 

947318 

424 

052682 

28 

33 

821693 

237 

874121 

187 

947572 

424 

052428 

27 

34 

821835 

237 

874009 

187 

947826 

424 

052174 

26 

35 

821977 

237 

873896 

187 

948081 

424 

051919 

25 

36 

822120 

237 

873784 

187 

948336 

424 

05U-64 

24 

37 

822262 

237 

873672 

187 

948590 

424 

051410 

23 

38 

822404 

237 

873560 

187 

948844 

424 

051156 

22 

39 

822546 

237 

873448 

187 

9VJ099 

424 

050901 

21 

40 

SZHffl 

236 

873335 

187 

949353 

424 

050647 

20 

41 

9.822SH) 

236 

9.873223 

187 

9.949607 

424 

10.050393 

19 

42 

822972 

236 

8731  10 

IPS 

949662 

424 

050138 

18 

43 

823114 

236 

872998 

188 

950116 

424 

049884 

17 

44 

823255 

236 

872885 

188 

950370 

424 

049630 

16 

45 

823397 

236 

872772 

188 

950625 

424 

049375 

15 

46 

823.539 

236 

872659 

188 

950879 

424 

049121 

14 

47 

823'680 

235 

872547 

188 

951133 

424 

048867 

13 

48 

335 

872434 

188 

951388 

424 

048612 

12 

49 

82.'?%3~ 

235 

872321 

188 

951642 

424 

048358 

11 

50 

824104 

235 

872208 

188 

951896 

424 

048104 

10 

51 

9.824245 

235 

9.872095 

189 

9-952150 

424 

10.047850 

9 

52 

824386 

235 

871981 

189 

952405 

424 

047595 

8 

53 

824527 

235 

871868 

189 

952659 

424 

047341 

7 

54 

824668 

234 

871755 

189 

952913 

424 

047087 

6 

55 

824808 

234 

871641 

189 

953167 

423 

046833 

5 

56 

824949 

234 

871528 

189 

953421 

423 

046579 

4 

57 

825090 

234 

871414 

189 

953675 

423 

046325 

3 

58 

8252:*0 

234 

871301 

189 

953929 

423 

046071 

2 

59 

825371 

234 

871187 

189 

954183 

423 

045817 

1 

60 

825511 

234 

871073 

190 

954437 

423 

045563 

0 

|   Cosine       |   Sine         Cotang.          Tang.   M.  [ 

48  Degrees. 


€0 


(42  Degrees.)     A  TABLR  or  LOGARITHMIC 


M.  j   Sine     D.   |  Cosine  |  D.    Tang.   |   D.   |  Cotang  f 

0 

9.825511 

234 

9,871073 

190 

9.054437 

423 

10.045563   60 

1 

825651 

233 

870960 

190 

954691 

423 

045309   59  ' 

2 

835791 

233 

870P46 

190 

954945 

4-23 

045055  i  58 

3 

825931 

233 

870732 

190 

955200 

423 

044800   57  » 

4 

826071 

233 

870618 

190 

•155454 

4-23 

044541  i   5t> 

5 

826211 

233 

870504 

190 

955707 

423 

044293  i  55 

6 

826351 

233 

870390 

190 

955961 

4-23 

044039   54 

7 

826491 

233 

870276 

190 

956215 

423 

043785   53 

8 

826631 

233 

870161 

190 

956469 

4-23 

013531   52 

9 

821.770 

232 

870047 

191 

958723 

423 

043-277   r,| 

10 

826910 

232 

869933 

191 

956977 

423 

043023 

50 

11 

9.827049 

232 

9.869818 

191 

9.957231 

423 

10.042709 

49 

12 

8-27189 

232 

869704 

191 

957485 

423 

042515 

48 

13 

827328 

232 

869589 

191 

957739 

423 

042261 

47  ! 

14 

827467 

232 

869474 

191 

957993 

423 

042007 

46 

15 

8-27606 

232 

869360 

KB 

958246 

423 

94J754 

45 

16 

8-27745 

232 

8H9245 

191 

958500 

423 

041500 

44 

17 

827884 

231 

869130 

191 

958754 

423 

041246 

43 

18 

828023 

231 

869015 

192' 

959008 

423 

040992 

42 

19 

828162 

231 

868900 

192 

959262 

423 

040738 

41 

20 

828301 

231 

868785 

192 

959516 

423 

040484 

40 

21 

9.828439 

231 

9.8G8670 

192 

9.959769 

423 

10.040231 

39 

22 

828578 

231 

868555 

192 

90-T023 

423 

039977 

3& 

23 

828716 

231 

868440 

192 

960277 

423 

039723 

37 

24 

8-28855 

230 

868324 

192 

960531 

423 

039469 

36 

25- 

828993 

230 

868209 

192 

960784 

423 

03&216 

35 

36 

829131 

230 

868093 

192 

961038 

423 

038962 

34 

27 

829269 

230 

867978 

193 

961291 

423 

038709 

33 

28 

829407 

230 

8(37802 

193 

9.'il545 

423 

038455 

32 

20 

829545 

230 

867747 

193 

961799 

423 

038201 

31 

30 

829683 

230 

867631 

193 

962052 

423 

037948 

30 

31 

9.829821 

229 

9.8SJ7515 

193 

9.9IW306 

423 

10,037694 

29 

32 

829959 

229 

867399 

193 

96-25()0 

423 

037440 

28 

33 

830097 

229 

867283 

193 

962813 

423 

037187 

27 

!  34 

830234 

229 

867167 

193 

»530G7 

423 

036933 

26 

35 

8.30372 

2-29 

867051 

193 

963320 

433 

036680 

25 

36 

830509 

229 

866935 

194 

963574 

423 

036426 

24 

37 

830646 

229 

866819 

194 

963827 

423 

036173 

23 

38 

830784 

229 

866703 

194 

964081 

423 

035919 

22 

39 

830931 

228 

866586 

194 

964335 

423 

035665 

21 

40 

831058 

228 

866470 

194 

964588 

422 

035412 

20 

41 

9.831195 

228 

9-866353 

194 

9.964842 

422 

10.035158 

19 

42 

831332 

228 

866237 

194 

965095 

422 

034905 

18 

43 

831469 

228 

866120 

194 

965349 

422 

034651 

17 

44 

831606 

228 

866004 

195 

965602 

422 

034398 

16 

45 

831742 

228 

865887 

195 

965855 

422 

034145 

15 

46 

831879 

228 

865770 

195 

966109 

422 

033891 

14 

47 

832015 

227 

865653 

195 

966362 

422 

033638 

13 

48 

832152 

227  ' 

865536 

195 

966616 

422 

033384 

12 

49 

832288 

227 

865419 

195 

966869 

422 

033131 

11 

50 

832425 

227 

865302 

195 

967123 

422 

032877 

10 

51 

9.832561 

227 

9-865185 

195 

9.907376 

422 

TO.  032624 

9 

52 

832697 

227 

865068 

195 

967629 

422 

032371 

8 

53 

832833 

227 

864950 

195 

967883 

422 

032117 

7 

54 

832969 

226 

864833 

196 

968136 

4-2-2 

031864 

6 

55 

833105 

226 

864716 

196 

968389 

422 

031611 

5 

56 

833241 

226 

864598 

196 

968643 

4,'-2 

031357 

4 

57 

833377 

226 

864481 

196 

968896 

422 

031104 

3 

58 

833512 

226 

864363 

196 

969149 

422 

030851 

2 

59 

833648 

226 

864245 

196 

969403 

422 

030597 

1 

60 

833783 

226 

864127 

196 

969656 

422 

030344 

0 

Cosine 

|   Sine 

]  Cotang.  |      [   Tang.   |  M.  1 

47  Degrees. 


SINES  AND  TANGENTS.     (43  Degrees.) 


61 


'  M.  |   Sine     D   |  Cosine   |  D.  |  Tang.     D.     Cotang.  | 

0 

9.833783 

226 

9.864127 

196 

9.969656 

422 

10.030:544 

60 

1 

833919 

22.5 

864010 

196 

90991)9 

422 

030091 

59 

2 

834054 

225 

863892 

197 

970  162 

422 

0298118 

58 

3 

834189 

225 

863774 

197 

970416 

422 

029584 

57 

4 

834325 

225 

863656 

197 

97000!) 

422 

029331 

56 

5 

834460 

225 

863538 

197 

970922 

422 

029078 

55 

0 

834595 

223 

8li3419 

197 

971175 

•1-22 

028825 

54 

7 

834730 

2-2.-> 

Hii^ioi 

197 

971429 

422 

028571 

53 

8 

834805 

823 

863183 

197 

971682 

422 

028318 

52 

9 

834999 

224 

863064 

197 

97l!):r, 

422 

028065 

51 

10 

835134 

224 

862946 

198 

972188 

422 

027812 

50 

11 

9.835269 

B24 

9.862827 

198 

9.972441 

422 

10.027.-,.-,!) 

49 

1-2 

835403 

224 

802709 

198 

972!  594 

422 

027306 

48 

13 

835538 

834 

862590 

198 

972948 

422 

027052 

47 

14 

835672 

224 

863471 

198 

973201 

422 

020799 

46 

15 

835807 

224 

862353 

198. 

973454 

422 

026546 

45 

16 

835941 

224 

862234 

198 

973707 

422 

020293 

44 

1? 

8:51)075 

223 

862115 

198 

973960 

422 

026040 

43 

18 

836209 

223 

861996 

198 

974213 

422 

025787 

42 

19 

836343 

223 

861877 

198 

974406 

422 

025534 

41 

20 

836477 

223 

861758 

199 

974719 

422 

025281 

40 

21 

9.8361)11 

223 

9-861638 

199 

9.974973 

422 

10.025027 

39 

22 

83(5745 

223 

861519 

199 

975226 

422 

0-24774 

38 

23 

836878 

823 

861400 

199 

975479 

422 

024521 

37 

24 

837012 

222 

861280 

199 

975732 

422 

024268 

36 

25 

837146 

2-22 

Hi;  1  Mil 

199 

975985 

422 

024015 

35 

26 

K5727!) 

222 

861041 

199 

!)702:58 

422 

023762 

34 

27 

837412 

222 

860922 

199 

976491 

422 

023509 

33 

28 

837546 

222 

860802 

199 

976744 

422 

023256 

32 

29 

837679 

222 

860682 

200 

976997 

422 

023003 

31 

30 

837812 

222 

860562 

200 

977250 

422 

022750 

30 

31 

9.8371)45 

2-22 

9-860442 

200 

9.977503 

422 

10.022497 

29 

32 

8:58078 

221 

860322 

200 

977755 

422 

022244 

28 

33 

838211 

221 

860202 

200 

9780»9 

422 

021991 

27 

34 

838344 

221 

860082 

200 

978262 

422 

021738 

26 

35 

838477 

221 

859962 

2l>0 

978515 

422 

021485 

25 

36 

8:58010 

221 

850842 

200 

978768 

422 

(12  1-2:5-2 

24 

37 

838742 

2-21 

8.19721 

201 

979021 

422 

020979 

23 

38 

838875 

221 

s.iinioi 

201 

979274 

422 

020726 

22 

39 

839007 

221 

859480 

201 

979527 

422 

020473 

21 

40 

839140 

220 

859360 

201 

979780 

422 

0-20-220 

20 

41 

9.839272 

220 

9.859239 

201 

9.930033 

422 

10.019967 

19 

42 

839404 

220 

859119 

201 

980286 

422 

019714 

18 

43 

Kl'.).->:50 

220 

858998 

2J1 

980538 

422 

019462 

17 

44 

839668 

220 

858877 

201 

980791 

421 

019209 

16 

45 

8.J9800 

220 

858756 

202 

981044 

421 

018956 

15 

46 

839932 

220 

858635 

202 

981297 

421 

018703 

14 

47 

840064 

219 

858514 

202 

981550 

421 

018450 

13 

48 

840191) 

219 

858393 

202 

981803 

421 

018197 

12 

49 

840328 

219 

858272 

202 

982056 

421 

0171)44 

11 

50 

840459 

219 

858151 

202 

982M9 

421 

017691 

10 

51 

9.840591 

219 

9.858029 

202 

9.982562 

421 

10.017438 

9 

.V2 

840722 

219 

857908 

202 

982814 

421 

017186 

8 

53 

840854 

219 

857786 

202 

983067 

421 

010933 

7 

54 

840985 

219 

857665 

203 

983320 

421 

016680 

6 

55 

841116 

218 

857543 

203 

983573 

421 

016427 

5 

56 

841247 

218 

857422 

203 

983826 

421 

016174 

4 

57 

841378 

218 

857300 

203 

984079 

421 

015921 

3 

58 

841509 

218 

857178 

203 

984331 

421 

015669 

2 

59 

841640 

218 

857056 

203 

984584 

421 

oi:,  no 

1 

60 

841771 

218 

856334 

203 

JW4K17 

421 

015163 

0 

|   Cosine       |   Sine   |     !  Cotang.  |         Tang.  |  M. 

46  Degrees. 


(44  Degrees.)    LOG.  SINES  AND  TANGENTS. 


M.    Sine     D.    Cosine    D.    Tang.     D.  |  Cotang. 

0 

9.841771 

218 

9.856934  1  203 

9.984837 

421 

10.015163 

60 

1 

841902 

218 

856812  1  203 

985090 

421 

014910 

59 

2 

842033 

218 

856690 

204 

985343 

421 

014657 

58 

3 

842163 

217 

856568 

204 

985596 

4'21 

014404 

57 

4 

842294 

217 

856446 

204 

985848 

421 

014152 

56 

5 

842424 

217 

856323 

204 

98lil01 

421 

013899 

55 

'  6 

842555 

217 

856201 

204 

98G354 

421 

013646 

54 

7 

842685 

217 

856078 

204 

98G607 

421 

013393 

53 

8 

842815 

217 

855956 

204 

986860 

421 

013140 

52 

9 

812946 

217 

855833 

204 

987112 

421 

012888 

51 

10 

843076 

217 

855711 

205 

987365 

421 

012635 

50 

11 

9.843206 

216 

9.855588 

205 

9.987618 

421 

10.012382 

49 

12 

843336 

216 

855465 

205 

987871 

421 

012129 

48 

13 

843466 

216 

855342 

205 

988123 

421 

011877 

47 

14 

843595 

216 

855219 

205 

988376 

421 

011624 

46 

15 

843725 

216 

855096 

205 

988629 

421 

011371 

45 

16 

843855 

216 

854973 

205 

988882 

421 

011118 

44 

17 

843984 

216 

854850 

205 

989134 

421 

010866 

43 

18 

844114 

215 

854727 

206 

989387 

421 

010613 

42 

19 

844243 

215 

854603 

206 

989640 

421 

010360 

41 

20 

844372 

215 

854480 

206 

989893 

421 

010107 

40 

21 

9.844502 

215 

9-854356 

206 

9-990145 

421 

10.009855 

39 

22 

844631 

215 

854233 

206 

990398 

421 

009602 

38 

23 

844760 

215 

854109 

206 

990651 

421 

009349 

37 

24 

844889 

215 

853986 

206 

990903 

421 

009097 

36 

25 

845018 

215 

853862 

206 

991156 

421 

008844 

35 

26 

845147 

215 

853738 

206 

991409 

421 

008591 

34 

27 

845276 

214 

853614 

207 

39J6R2 

421 

008338 

33 

28 

845405 

214 

853490 

207 

991914 

421 

00808(5 

32 

29 

845533 

214 

853366 

207 

99-2167 

421 

007833 

31 

30 

845662 

214 

853242 

207 

992420 

421 

007580 

30 

31 

9.845790 

214 

9.853118 

207 

9-992672 

421 

10.007328 

29 

32 

845919 

214 

852994 

207 

992925 

421 

007075 

28 

33 

846047 

214 

852869 

207 

993178 

421 

006822 

27 

34 

846175 

214 

852745 

207 

993430 

421 

OOG570 

26 

35 

846304 

214 

852620 

207 

993683 

421 

006317 

25 

36 

846432 

213 

852496 

208 

993936 

421 

000064 

24 

37 

846560 

213 

852371 

208 

994189 

421 

005811 

23 

38 

846688 

213 

852247 

208 

994441 

421 

005559 

22 

39 

846816 

213 

852122 

208 

994694 

421 

005306 

21 

40 

846944 

213 

851997 

208 

994947 

421 

005053 

20 

41 

9-847071 

213 

9-851872 

208 

9-995199 

421 

10.004801 

19 

42 

847199 

213 

851747 

208 

995452 

421 

004548 

18 

43 

847327 

213 

851622 

208 

995705 

421 

004295 

17 

44 

847454 

212 

851497 

209 

995957 

421 

004043 

16 

45 

847582 

212 

851372 

209 

996210 

421 

003790 

15 

46 

847709 

212 

851246 

209 

996463 

421 

003537 

14 

47 

847836 

212 

851121 

209 

996715 

421 

003285 

13 

48 

847964 

212 

850996 

209 

996968 

421 

003032 

12 

49 

848091 

212 

850870 

209 

997221 

421 

002779 

11 

50 

848218 

212 

850745 

209 

997473 

421 

002527 

10 

51 

9-848345 

212 

9.850619 

209 

9.997726 

421 

10.002274 

9 

52 

848-172 

211 

850493 

210 

997979 

421 

002021 

8 

53 

848599 

211 

850368 

210 

998231 

421 

001769 

7 

54 

848726 

211 

850242 

210 

998484 

421 

001516 

6 

55 

848852 

211 

850116 

210 

998737 

421 

001263 

5 

56 

848979 

211 

849990 

210 

998989 

421 

001011 

4 

57 

849106 

211 

849864 

210 

999242 

421 

000758 

3 

58 

849232 

211 

849738 

210 

999495 

421 

000505 

2 

59 

849359 

211 

849611 

210 

999748 

421 

000253 

1 

60 

849485 

211 

849485 

210 

10000000 

421 

000000 

0 

|  Cosine          Sine        Cotang.       |   Tang.  |  M. 

45  Degrees. 


TABLE 

OP 

NATURAL  SINES  AND  TANGENTS; 

TO 

EVERY  TEN  MINUTES  OF  A  DEGREE. 


IF  the  given  angle  is  less  than  45°,  look  for  the  title  of  the  column,  at 
the  top  of  the  page ;  and  for  the  degrees  and  minutes,  on  the  left.  But 
if  the  angle  is  between  45°  and  90°,  look  for  the  title  of  the  column,  at 
the  bottom;  and  for  the  degrees  and  minutes  on  the  right. 

The  Secants  and  Cosecants,  which  are  not  inserted  in  this  table,  may 
be  easily  supplied.  If  1  be  divided  by  the  cosine  of  an  arc,  the  quotient 
will  be  the  secant  of  that  arc.  (Art.  228.)  And  if  1  be  divided  by  the 
sine,  the  quotient  will  be  the  cosecant. 


NATURAL  SINKS. 


0  Deg. 

1  Deg. 

2  Deg. 

3  Deg. 

4  Deg. 

Nat. 

N.Co 

Nat.  iN.Co 

I  Nat. 

N.Co 

~NHtT 

N.  Co- 

Nat, 

N.  Co- 

M 

Sine 

Sine 

Sine.  |  Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

Sine 

.M 

~o 

OJOOO 

Unit. 

01745 

991)85 

"03490 

99939 

05234 

998(  3 

00970 

99756 

(50 

1 

00029 

00000 

01774 

99984 

03519 

99938 

05263 

99801 

07005 

99754 

59 

2 

00053 

(10000 

01803 

99984 

03548 

99937 

05292 

99861) 

07034 

99752 

58 

3 

00087 

00000 

01832 

99983 

o:?577 

99930 

05321 

99858 

07003 

99750 

57 

4 

00116 

00000 

018tJ2 

99983 

03606 

99935 

0.1350 

99857 

07092 

H9748 

56 

5 

00145 

00000 

01891 

99982 

03635 

99934 

05379 

99855 

07121 

99746 

55 

6 

00175 

00000 

01920 

99982 

03664 

99933 

05408 

9J854 

07150 

99744 

54 

7 

00204 

00000 

01949 

99981 

03693 

99932 

05437 

99852 

07179 

99742 

53 

8 

00233 

00000 

01978 

99980 

03723 

99931 

05406 

9985J 

07208 

1)9740 

52 

9 

00262 

00000 

02007 

99980 

03752 

99930 

05495 

99849 

07237 

99738 

51 

10 

00291 

00000 

0-2030 

99979 

03781 

1)992!  1 

05124 

99847 

07366 

99730 

50 

1] 

0032!) 

99999 

02065 

99979 

03810 

99927 

05553 

99840 

07295 

99734 

49 

18 

00349 

99999 

02094 

99978 

03839 

99920 

05582 

99844 

07324 

99731 

48 

13 

00378 

99999 

02123 

99977 

03868 

99925 

05611 

99842 

07353 

99729 

47 

14 

00407 

99999 

02152 

99977 

0:^897 

9J924 

05040 

99841 

07382 

99727 

46 

15 

00436 

99999 

02181 

99976 

03926 

99923 

05069 

99839 

07411 

99725 

45 

16 

00405 

99999 

02211 

99976 

03955 

99922 

05098 

99838 

07440 

99723 

44 

17 

00495 

99999 

OJ240 

99975 

03984 

999-21 

05727 

998H6 

07469 

99721 

43 

18 

005-24 

99999 

02209 

99974 

04013 

99919 

05756 

99834 

07498 

99719 

42 

19 

01)553 

99998 

02298 

99974 

04042 

93918 

05785 

99S33 

075-27 

99710 

41 

20 

00582 

99998 

02327 

99973 

04071 

939  17 

05814 

99831 

0755(i 

99714 

40 

21 

00611 

99998 

02356 

99972 

04100 

99:)10 

05844 

9982;) 

07585 

99712 

39 

2-2 

001)40 

99998 

02385 

99972 

04129 

09915 

05873 

99827 

07014 

99710 

38 

23 

00660 

99998 

02414 

99971 

0*138 

99913 

OSMSZ 

99826 

07043 

99708 

37 

24 

00698 

99998 

02443 

99970 

04188 

99912 

05931 

99824 

07672 

5*9705 

36 

25 

007-27 

99997 

02472 

99909 

04217 

99911 

05960 

9:)822 

07701 

99703 

35 

26 

00756 

99997 

02501 

99969 

04246 

99910 

05989 

99821 

07730 

99701 

34 

27 

00785 

99997 

02531) 

99968 

04275 

99909 

06018 

99819 

07759 

J9699 

33 

28 

00814 

99997 

025(50 

99967 

04904 

99907 

00047 

9J8I7 

07788 

99090 

32 

29 

00844 

99996 

02589 

99960 

04333 

99900 

06076 

99815 

07817 

19094 

31 

30 

00873 

99996 

02618 

99966 

04362 

J9905 

06105 

99S13 

07846 

99692 

30 

31 

00902 

99996 

02647 

999R5 

04391 

99904 

06134 

99812 

07875 

99689 

29 

32 

00931 

99990 

02676 

99904 

044-20 

99902 

08M3 

91)814) 

07904 

J9087 

28 

33 

00960 

99995 

02705 

99903 

04449 

99001 

00192 

99803 

07933 

J9085 

27 

34 

00989 

99995 

02734 

99963 

04478 

)990i) 

00-221 

99801) 

07962 

J9083 

20 

35 

01018 

99995 

02763 

99902 

04507 

)9898 

06250 

99804 

07991 

J9080 

25 

36 

01047 

99995 

02792 

99961 

04536 

99897 

06279 

99803 

08020 

99678 

24 

37 

01076 

99994 

02821 

99900 

04505 

.»989!i 

06308 

99801 

08049 

99676 

23 

38 

01105 

99994 

0-2850 

99959 

04594 

99894 

06337 

99799 

08078 

91)073 

22 

39 

01134 

99994 

02379 

99959 

04623 

99893 

06306 

99797 

08107 

99671 

21 

40 

01164 

99993 

0-2908 

99958 

04653 

)9892 

OG395 

99795 

03130 

J9fifi8 

20 

41 

01193 

99993 

02938 

99957 

-04682  ' 

)9890 

00424 

99793 

08105 

99600 

19 

42 

01222 

99993 

02967 

99950 

04711 

99889 

06453 

9971)2 

08194 

)!)004 

18 

43 

01-251 

99992 

02996 

99955 

04740 

99888 

00482 

99790 

08223 

99061 

17 

44 

01280 

99992 

03025 

99954 

04769 

J9886 

00511 

99788 

08252 

99059 

16 

45 

01309 

99991 

03054 

99953 

04798 

J9885 

06540 

99786 

08281 

99057 

15 

46 

01338 

99991 

03083 

99952 

04827 

99883 

06569 

99784 

OH310 

99654 

14 

47 

01367 

99991 

03112 

99932 

04850 

99832 

06598 

99782 

08339 

J9052 

13 

48 

01396 

99990 

03141 

9995  J 

G4885 

)9881 

06627 

99780 

08308 

)9049 

12 

49 

01425 

99990 

03170 

9995:) 

04914 

)987l) 

06656 

99778 

08397 

99647 

11 

50 

01454 

99989 

03199 

99949 

04943 

99878 

06085 

99776 

0842(5 

)9044 

10 

51 

01483 

99989 

03-228 

99948 

04972 

)9870 

06714 

99774 

08455 

)9042 

9 

52 

01513 

99989 

03-257 

99947 

05001 

99875 

06743 

99772 

08484 

99039 

8 

53 

1542 

99988 

03280 

99940 

05030 

J9873 

00773 

99770 

08513 

(9037 

7 

54 

01571 

99938 

03316 

99945 

05059 

Q9'«72 

OH802 

99708 

OF5  12 

99035 

6 

55 

01600 

99987 

03345 

99944 

05088 

IDH70 

00831 

99760 

08571 

)9C>32 

5 

56 

01029 

99937 

03374 

99913 

05117 

I'Hfi'.l 

06860 

99704 

08(500 

J9030 

4 

57 

Olfi58 

99986 

03403 

99942 

05140 

MH-r; 

00889 

99702 

08639 

19027 

3 

58 

011)87 

99986 

03432 

99911 

05175 

99806 

00918 

99760 

08058 

19625 

2 

59 

01716 

99985 

03461 

99940 

05205 

99304 

00947 

99758 

08687 

J9622 

1 

"M" 

N.CS. 

N.  S. 

N.08. 

N.S. 

N.  CS. 

N.S. 

N.CS. 

N.S. 

\  CS.|  N  S. 

M 

89  Deg. 

88  Deg. 

87  Deg. 

86  Deg. 

85  Deg. 

NATURAL    SINKS. 


5  Deg. 

6  Deg. 

7  Deg. 

8  Deg.    9  Deg. 

M 

N.a 

N.CS. 

N.S. 

\  .(>« 

N.S.  |N.CS 

i\.S. 

N.CS.  N.S. 

N.CS 

M 

~0 

08716 

99619 

10453 

99452 

:  99255 

13917 

99027  15643 

987(»9 

00 

1 

08745 

99017 

10482 

SI1M41) 

12216 

99251 

13946 

99023  1  51  172 

98764 

59 

0 

08774 

!»:iiil4 

10511 

99441) 

12245 

H!)24.- 

13975 

99014)  15701 

98709 

58. 

3 

99!)  12 

10540 

9.)443 

12274 

99J44 

14004 

99015  1573H 

98755 

57 

4 

08831 

99009 

L0568 

99440 

123(12 

D'KMO 

14033 

99011  !.->;.> 

98751 

56 

5 

08866 

SI9i;.)7 

10597 

ii;M37 

1233] 

99237 

14061 

H'.MKM;  15787 

98740 

.")") 

6 

08889 

99604 

10626 

9;)4:!4 

12360 

99233 

14090 

99002   158  1C 

98741 

54 

7 

08918 

<);>oo2 

I  ill,  .M 

99431 

[2389 

!);I230 

Hllll 

9*!)<is  J;V!5 

98737 

53 

8 

08947 

<);).->:t;i 

10084 

99428 

12418 

99220 

14148 

1KIHI   15H-I3 

98732 

52 

9 

08-J76 

99."i:»:i 

10713 

1)9124 

12447 

99222 

14177 

989!»0  lf)!)02 

98728 

.")! 

10 

09009 

99594 

10742 

H942I 

12470 

»1»21«) 

14205 

1593] 

98723 

50 

11 

09034 

9:»591 

11*771 

!»»4IH 

1  25114 

99215 

14234 

98982  159.-)!) 

98718 

4!) 

IS 

09063 

99588 

10800 

11:141.-) 

12533 

9921  1 

14263 

!)'-'.)•;  8  15988 

9H'  14 

48 

13 

09092 

99586 

10839 

1)1)4  1  2 

12562 

1)9208 

1421)2 

si«)7:j  Kioi: 

98709 

47 

14 

09121 

99583 

10858 

91)409 

12591 

99204 

14320 

98909  1004() 

98704 

48 

IS 

09150 

99580 

10887 

99406 

12620 

99200 

14349. 

98i)65  10074 

98700 

45. 

10 

09179 

99578 

L0916 

!):)102 

12640 

99197 

14378 

98961  i  16103 

98695  1  44 

17 

09208 

99575 

10945 

99399 

12878 

99193 

14407 

98957  10132 

98(590  43 

18 

(U237 

99572 

10973 

!):>:ill>; 

12706 

99189 

14436 

l)^!l.-.3  16160 

98086  42 

19 

09266 

99570 

11002 

!I9393 

12735 

99186 

14464 

98948  1  16189 

98681  '  41 

20 

09295 

D9567 

11031 

99390 

12764 

99  i>2 

14493 

98944   Hi21f 

98076  ;  40 

21 

09324 

U9.-X54 

11060 

99386 

12793 

99178 

14522 

98940  16240 

98671 

39 

2-2 

09353 

!I9.->I>2 

11089 

99383 

12822 

99175 

14551 

98936 

16275 

98667 

38 

23 

09382 

99559 

11118 

DittrfO 

12851 

99171 

14580 

98931 

10304 

98662 

37 

21 

09411 

99556 

11147 

SW377 

12880 

99167 

14608 

1)8927 

16333 

98657 

3(5 

25 

0:)440 

99553 

11176 

99374 

121)08 

'.iiiua 

14637 

98923 

16361 

98652 

35 

20 

09469 

99551 

11205 

JU370 

121)37 

0il60 

14666 

98919  16390 

98648 

34 

27 

09498 

99548 

1  1234 

99367 

12966 

9915(5 

14695 

9H914  1041!) 

98643 

33 

2d 

09527 

99545 

11283 

993G4 

12995 

99152 

14723 

98910  10M7 

1)9(538 

32 

2'J 

09556 

99542 

11291 

9930:1 

131)24 

99148 

14752 

98906  104:0 

98033 

31 

30 

09585 

99540 

11320 

9J357 

13053 

99144 

14781 

98902 

16505 

98629 

30 

31 

09G14 

99537 

11349 

99354 

13081 

1)9141 

14810 

98897 

16533 

98624 

29 

32 

09042 

99534 

11378 

1)!»35I 

13110 

«9137 

14838 

98893  10502 

98619 

28 

33 

09671 

99531 

114(17 

9;)347 

13139 

99133 

14867 

98889  16591 

98G14 

27 

31 

09700 

U9528 

11436 

99344 

13168 

99129 

14896 

98884  16020 

98609 

26 

3.5 

0972a 

:i:».VJO 

L146i 

9.t:;n 

13197 

99125 

14925 

98880  1004.-' 

1)8(504 

25 

36 

09758 

99523 

L1494 

99337 

13226 

'.19122 

14954 

98876  |  16677 

08600 

24 

37 

09787 

99520 

11523 

99:n4 

13254 

14982 

i)887l  I  16706 

96595 

23 

38 

09816 

99517 

11552 

99331 

13283 

99114 

15011 

)rW,7  10731 

18590 

22 

39 

09845 

99514 

11580 

9<>327 

13312 

99110 

15640 

10703 

18585 

21 

40 

OJ874 

9951  1 

11609 

99324 

13341 

1)1)106 

15069 

)8858  107'.)2 

18580 

BO 

41 

Od903 

9951)8 

11638 

13370 

95)102 

15097 

«as  i 

98575 

19 

42 

09932 

99506 

U667 

99317 

13399 

99098 

15121) 

98849 

16849 

98570 

JH 

43 

09961 

9951)3 

11696 

99314- 

13427 

l)i)l)l»4 

15155 

)8845 

16878 

98965 

17 

44 

09990 

99509 

L1725 

99310 

1345G 

99091 

15184 

98841 

16996 

IIH.IOI 

10 

45 

10019 

99497 

11754 

99307 

13485 

99087 

15212 

J883G 

16935 

98556 

15 

40 

10048 

99494 

11783 

99303 

13514 

99083 

15241 

98832 

16964 

98551 

14 

47 

10077 

99491 

11812 

99300 

13543 

991)79 

15270 

98827 

10992 

98546 

13 

48 

10106 

99  188 

L1840 

99297 

13572 

99075 

15292 

98fl23 

17021 

9ar)41 

12 

49 

10135 

99485 

11869 

99293 

13600 

99071 

15327 

9d818 

17050 

98536 

11 

50 

10164 

99482 

11898 

1I-I2S),') 

13629 

99067 

1535(5 

J8814 

17078 

88931 

10 

51 

10192 

99479 

1  1927 

99-286 

13658 

99063 

15385 

98809 

17107 

98996 

9 

52 

1022] 

9947(5 

11956 

99283 

13687 

99059 

15414 

18805 

1713G 

98521 

8 

53 

J0250 

99473 

11985 

99279 

13716 

99055 

15442 

18800 

17164 

98516 

7 

54 

10279 

99470 

12014 

99276 

13744 

99051 

15471 

>-;;H; 

17193 

98511 

6 

55 

1030.-! 

99467 

12043 

99272 

13773 

99047 

15500 

J8791 

I72S2 

98508 

5 

56 

10337 

994(54 

12071 

99269 

13802 

991)43 

15529 

18787 

17250 

1)8501 

4 

57 

10386 

1U4G1 

12100 

99265 

13831 

1)9039 

15557 

172'<1) 

9-41)0 

3 

58 

10395 

99458 

12129 

99262 

13860 

99035 

15586 

)--,  >' 

17308 

98491 

2 

59 

10424 

99455 

12158 

99258 

13889 

99031 

i.-.iiir. 

98773 

1733(5 

9848B 

1 

M~ 

N.CU 

N.S. 

N.CS.I  N  S 

N.CS 

N.S. 

v.cs 

N.S. 

N.CS. 

N.S. 

~»T 

84  Deg. 

83  Deg. 

"WDegT 

81  Deg. 

~80~Deg. 

66 


NATURAL    SINES. 


M 

10  Deg. 

11  Deg. 

12  Deg. 

13  Deg.    14  Deg. 

M 

N.S. 

N.CS 

N.S. 

N.CS 

N.S. 

N.O 

N.S.  IN.CS.  N.S. 

N.  US 

0 

17305 

98481 

19081 

9dn>: 

20791 

97815 

22495  97437  swiUi. 

97030 

60 

1 

17393 

98476 

19109 

98157 

20820 

97809 

22523 

97430  2-12"2( 

97023 

59 

<2 

17422 

98471 

19138 

9815$. 

20848 

9780: 

2-255. 

97424  24241 

97015 

58 

3 

17451 

98466 

19167 

98146 

20877 

9779" 

22580 

97417  24277 

97008 

57 

4 

17479 

98461 

19195 

98140 

20905 

97791 

22(508 

97411  24305 

97001 

56 

e 

17508 

98455 

19224 

98135 

j  20933 

9778- 

22637 

97404  24333 

9C994 

55 

6 

17537 

98450 

19252 

98123 

20962 

97778 

22605 

97398  24302 

90987 

54 

•; 

17565 

98445 

19281 

98124 

2(1!  (UO 

97772 

:  22<5!»: 

97391  24390 

90980 

53 

8 

17594 

98440 

19309 

98118 

21019 

9776( 

22722 

97384  244  IS 

96973 

52 

g 

17623 

98435 

19338 

98112 

21047 

97700 

22750 

97378  244,46 

90900 

51 

10 

17651 

98430 

19366 

98107 

21076 

97754 

22778 

97371  -"447-. 

90959 

50 

11 

17680 

98425 

19395 

98101 

21104 

97748 

!  22807 

97305  24503 

915952 

49 

12 

17708 

98420 

19423 

9809P 

21132 

97742 

22835 

97358  24531 

901)45 

48 

13 

17737 

98414 

;  19452 

98090 

21101 

97735 

22803 

97:i.3l  24559 

96937 

47 

14 

17766 

98409 

19481 

98084 

21189 

977-29 

22892 

97345  24587 

96930 

40 

15 

17794 

98404 

19509 

98079 

21218 

97723 

22920 

'.17338  24015 

96923 

45 

16 

17823 

98399 

19538 

98073 

21246 

97717 

22948 

97331  24044 

96910 

44 

17 

17852 

98394 

19566 

98067 

21275 

9771  J 

22977 

973-25  24072 

90909 

43 

18 

17880 

98389 

19595 

OPflfil 

21303 

97705 

23005 

97318  247  OU 

90902 

42 

19 

17909 

98383 

19(523 

9805f 

21331 

97098 

23033 

97311  247-28 

90894 

41 

20 

17937 

98378 

19652 

98050 

21360 

97092 

'23002 

97304  24750 

96887 

40 

24 

17966 

98373 

19680 

98044 

21388 

9708f 

23090 

97298  24784 

90880 

39 

22 

17995 

98368 

19709 

98039 

21417 

97080 

23118 

97291  24813 

9C873 

38 

23 

18033 

98362 

19737 

98033 

21445 

97673 

23146 

97284  :  24841 

90806 

37 

24 

18052 

98357 

19766 

98027 

21474 

97667 

23175 

97278  24809 

90858 

36 

25 

J8081 

98352 

19794 

98021 

21502 

97601 

23203 

97271  24897 

90851 

35 

26 

18109 

98347 

19823 

98016 

21530 

97655 

23231 

97204  24925 

90844 

34 

27 

18138 

98341 

19851 

98010 

21559 

97648 

23260 

97257  24953 

96837 

33 

28 

18166 

98336 

19880 

98004 

21587 

97642 

23288 

97251  24982 

)(i829 

32 

29 

18195 

98331 

1990S 

97998 

21616 

97636 

23316 

97244  25010 

JC:8-2-2 

31 

30 

18224 

98325 

19937 

97992 

21  044 

97630 

23345 

97237  25038 

J0815 

30 

31 

18252 

98320 

19965 

97987 

21672 

97623 

23373 

97230  25066 

96807 

2s> 

32 

18281 

98315 

19994 

97981 

21701 

97617 

23401 

97223  25094 

96800 

28 

33 

18309 

98310 

200-2-2 

97975 

21729 

97611 

23429 

97217  25122 

96793 

27 

34 

18338 

96304 

20051 

979R9 

21758 

97604 

23458 

97-11)  25151 

JG780 

20 

35 

18367 

98299 

20079 

07903 

21786 

97598 

23486 

97203  |  25179 

96778 

25 

36 

18395 

98294 

20108 

97958 

21814 

97592 

23514 

97196  j  25207 

J6771 

24 

37 

18424 

98288 

20136 

97952 

21843 

97585 

23542 

97189  J25235 

J0764 

23 

38 

18452 

98283 

20165 

97946 

21871 

97579 

23571 

97182 

25263 

J0756 

22 

39 

18481 

98277 

20193 

97910 

21899 

97573 

23599 

97176 

25291 

J0749 

21 

40 

18509 

98272 

20222 

97934 

21928 

97566 

23027 

97169 

25320 

90742 

20 

41 

18538 

98267 

20250 

979-28 

21956 

97500 

23056 

97162 

25348 

90734 

19 

42 

18567 

98261 

20279 

979-22 

21985 

97553 

23684 

97155 

25376 

JG727 

18 

43 

18595 

98256 

20307 

97916 

22013 

97547 

23712 

97148 

25404 

96719 

17 

44 

18024 

98250 

20336 

97910 

22041 

97541 

23740 

97141 

25432 

J6712 

16 

45 

18652 

98245 

20364 

97905 

22070 

97534 

23769 

97134 

25460 

96705 

15 

46 

18681 

98240 

20393 

97899 

22098 

97528 

23797 

97127 

25488 

96697 

14 

47 

18710 

98234 

20421 

»7S93 

2-21-20 

97521 

23825 

97120 

2551(5 

)069() 

13 

48 

18738 

J8229 

20450 

J7887 

22155 

97515 

23853 

97113 

25545 

96682 

12 

49 

18767 

J8223 

20478 

97881 

22183 

97508 

23882 

97106 

25573 

J0075 

11 

50 

18795 

98218 

20507 

97875 

2-2212 

9750-2 

2r,910 

97100 

25601 

9(5007 

10 

51 

18824 

98212 

20535 

978(59 

22240 

9749(5 

23938 

97093 

25629 

96(500 

9 

52 

18852 

98207 

20563 

97863 

2-2208 

97489 

229H6 

97086 

25657 

90653 

8 

53 

18881 

J8201 

20592 

97857 

22297 

97483 

2.1995 

97079 

25685 

96(545 

54 

18910 

98196 

20620 

97851 

22325 

97470 

24023 

(7072 

25713 

96638 

6 

55 

18938 

J8190 

20649 

97845 

J2353 

)7470 

21051 

97005 

25741 

96030 

5 

56 

18967 

)8185 

20677 

)78:)9 

22382 

J7403 

24079 

97058 

25769 

96023 

4 

57 

18995 

98179 

20706 

17833 

22410 

>7457 

24108 

97051 

25798 

16615 

3 

58 

19024 

98174 

20734 

978-27 

2-243H 

)7450 

24136 

97044 

25826 

90008 

<2 

59 

19052 

98168 

20763 

97821 

22467 

97444 

24164 

97037 

25854 

J6600 

1 

M~ 

N.CS. 

N.S. 

N.CS.  [N.S. 

N.CS. 

N.S. 

N.CS. 

N.S. 

N.CS. 

N.S   M 

79  Deg. 

78  Deg. 

77  Deg. 

76  Deg. 

75  Deg.  i 

NATURAL  SINES. 


15  Deg. 

16  Deg. 

17  Deg. 

18  Deg. 

1  19  Deg. 

M 

N.CS. 

N.S.  I  N.CS. 

N.S. 

N.CS. 

N.S. 

N.CS. 

1  N.S. 

N.CS. 

M 

o 

25882 

96593 

27564  96126 

29237 

95630 

309  J2 

95106 

32557 

94552 

60 

1 

45910 

!«;.vc> 

27592 

96118 

39365 

95(522 

30929 

951  1!  ',7 

33584 

94542 

59 

35938 

96578 

27620 

96110 

.29293 

95613 

30957 

950.S8 

89612 

94533 

58 

3 

25986 

96570 

27648 

!»!>  102 

•29321 

95605 

30985 

95079 

33639 

94523 

57 

4 

25994 

965152 

-2711715 

96094 

39348 

95596 

31012 

95070 

38667 

94514 

5(5 

s 

20022 

9(5555 

27704 

96086 

29376 

95588 

31040 

95061 

32(594 

94504 

55 

(i 

38050 

9654? 

27731 

96078 

39404 

95579 

31068 

SK5052 

327-22 

944:15 

54 

38079 

9(5540 

27759 

9(5070 

29432 

95571 

31095 

95043 

3274!) 

1)4485 

53 

8 

20107 

915532 

27787 

ytii>i»2 

39460 

95562 

31123 

95033 

32777 

94476 

52 

9 

26135 

96524 

27815 

96054 

39487 

95554 

31151 

95024 

32804 

94166 

51 

10 

26163 

96517 

27843 

96046 

•211515 

95545 

31178 

95015 

32832 

94457 

50 

11 

3619] 

96509 

27871 

96037 

39543 

95536 

31206 

95006 

32859 

94447 

49 

19 

36219 

1)155(1-2 

27899 

9(50-2;> 

89571 

95528 

31233 

94997 

32887 

94438 

48 

13 

•20-247 

96494 

37997 

uiiir-21 

39599 

95519 

31-261 

94988 

32914 

!)H-2r> 

47 

14 

36375 

96486 

27955 

96013 

29626 

95511 

31289 

94979 

32942 

94418 

46 

15 

26303 

96479 

27983 

96005 

29654 

95502 

31316 

94970 

32969 

94409 

45 

16 

26331 

96471 

28011 

95997 

29682 

95493 

31344 

94961 

32997 

94399 

44 

17 

36359 

96463 

28039 

951)39 

29710 

95485 

31372 

94952 

33024 

94390 

43 

18 

36387 

96456 

28067 

95981 

29737 

95476 

31399 

94943 

33051 

91380 

42 

19 

86415 

96448 

28095 

95972 

29765 

95467 

31427 

94933 

33079 

94370 

41 

20 

36443 

96440 

28183 

95964 

29793 

95459 

31454 

94924 

33106 

94361 

40 

21 

26471 

96433 

38150 

95956 

29821 

95450 

3J482 

94915 

33134 

94351 

39 

2-2 

26500 

96425 

28178 

95948 

29849 

95441 

31510 

94906 

33161 

94342 

38 

23 

20.V28 

96417 

28206 

95940 

29876 

95433 

31537 

94897 

33189 

94332 

37 

24 

36556 

96410 

28234 

95931 

29904 

D.-.424 

315(55 

94888 

33216 

1)13-22 

36 

23 

265.-  1 

96402 

2-262 

95923 

291)3-2 

95415 

31593 

94878 

33244 

94313 

35 

2(5 

26612 

96394 

38290 

95915 

39960 

95407 

31620 

94869 

33271 

94303 

34 

27 

36640 

9(«86 

38318 

95:107 

29987 

95398 

31648 

948(50 

33898 

94293 

33 

28 

36668 

96379 

28346 

95?\),S 

30015 

95:J89 

31675 

94851 

33326* 

94284 

32 

29 

36696 

96371 

28374 

95890 

30013 

95380 

31703 

94842 

33353 

94274 

31 

30 

26724 

96363 

28402 

95882 

30071 

95372 

31730 

94832 

33381 

94264 

30 

31 

26752 

90355 

28429 

95874 

30098 

95360 

31758 

94823 

33408 

94254 

29 

32 

28780 

96347 

28457 

95865 

30126 

95354 

31786 

94814 

33436 

94245 

28 

33 

26808 

96340 

28485 

95857 

30154 

95345 

31813 

94805 

33463 

94235 

27 

34 

2683(5 

96332 

28513 

95849 

30182 

95337 

31841 

94795 

33490 

94225 

2(5 

35 

2(5864 

9(5324 

28541 

95841 

302011 

95*28 

31868 

94786 

33518 

94215 

25 

3f> 

36893 

9(531(5 

28569 

95832 

30237 

95319 

31896 

94777 

33545 

94206 

24 

37 

26920 

96308 

38597 

95824 

30265 

95310 

31923 

947(58 

33573 

94196 

23 

38 

26948 

96301 

38625 

95816 

30-2!'.; 

95301 

31951 

94758 

33600 

94186 

22 

39 

36976 

9(5293 

38652 

95807 

30320 

95293 

31979 

94749 

33627 

94176 

21 

40 

27004 

96285 

38680 

95799 

30348 

95-284 

3-200(5 

94740 

33655 

94167 

20 

41 

27032 

96277 

28708 

95791 

30376 

95-275 

3-2034 

84730 

33682 

94157 

19 

42 

27060 

9(5269 

38736 

95783 

30403 

9526(5 

32061 

94721 

33710 

94147 

18 

43 

27088 

96261 

28764 

95774 

30431 

95257 

32089 

94712 

33737 

94137 

17 

44 

27116 

9(5253 

28792 

9576(5 

30459 

95248 

32116 

94702 

33764 

94127 

16 

4-5 

27144 

96246 

28820 

95757 

30486 

95240 

32144 

94693 

33792 

U4118 

15 

46 

27172 

96238 

28847 

95749 

30514 

95231 

32171 

94684 

33819 

94108 

14 

47 

37300 

9(5230 

28875 

95740 

30542 

95222 

32199 

94674 

33846 

94098 

13 

48 

37938 

9(522-2 

28903 

1)5732 

30570 

95213 

3-2-227 

U4t;r>5 

33874 

9408H 

12 

49 

272.")(5 

96214 

28931 

95724 

30597 

95204 

32254 

94656 

33901 

94078 

11 

5il 

•27-28  I 

9(520(5 

38959 

95715 

30625' 

95195 

32282 

94646 

33089 

940(58 

10 

51 

273)2 

96198 

28987 

95707 

30653 

9518(5 

32309 

94637 

33956 

94058 

9 

52 

27340 

9(5  19i) 

39015 

95698 

30680 

95177 

32337 

94(527 

33!).-:) 

94049 

8 

53 

37368 

96182 

29042 

95690 

30708 

95168 

38364 

94618 

34011 

94039 

7 

54 

2739(5 

96174 

29;  170 

95631 

30736 

95159 

323112 

94(509 

34038 

1)10-29 

6 

55 

27424 

9'inii; 

39098 

95(573 

30763 

95150 

33419 

9459!) 

34065 

94019 

5 

56 

27452 

96158 

29136 

«j.-»r.»ii 

30791 

95142 

32-147 

94590 

34093 

94009 

4 

57 

37480 

96i.-)il 

39154 

95155(5 

30819 

95133 

32474 

94580 

34120 

93999 

3 

58 

27508 

1115142 

89182 

95617 

30846 

95121 

32503 

94571 

34147 

939S9 

2 

5'J 

*  2  7536 

96134 

29209 

95639 

30874 

95115 

32529 

94561 

34175 

93979 

1 

M 

N.CS. 

NTs. 

N.CS. 

'N.S. 

N.CS 

N.S. 

N.CS. 

N.S. 

N.CS. 

N.S. 

IT 

74  Deg. 

"73  Deg. 

72  Deg. 

71  Deg. 

70  Deg. 

NATURAL   SINES. 


20  Deg. 

21  Deg. 

22  Deg. 

23  Deg. 

'  24  Deg. 

M 

N.S. 

N.CS. 

iN.S.  IN.C6 

N.S. 

N.CS. 

N.S. 

N.CS 

i  N.  S. 

N.  OS 

M 

~0 

34-20-2 

93969 

35837  9J358 

374151 

92718 

39073 

92050 

40674 

91355 

60 

1 

3422J 

93959 

35804 

93348 

37488 

927U7 

39100 

92039 

40700 

91343 

59 

2 

34257 

93949 

35891 

93331 

37515 

92697 

39127 

92028 

40727 

91331 

58 

3 

34284 

93939 

35918 

933-27 

37542 

92686 

39153 

92016 

40753 

91319 

57 

4 

34311 

93929 

35945 

93316 

37569 

92675 

39180 

92005 

40780 

91307 

56 

5 

34339 

93919 

35973 

93306 

37595 

92664 

39207 

91994 

40806 

91295 

55 

6 

34366 

93909 

36000 

93295 

37622 

92653 

39234 

91982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

91971 

40860 

91272 

53 

8 

34421 

93889 

36054 

93274 

37676 

92631 

39287 

91959 

40886 

91260 

52 

0 

34448 

93879 

36081 

93264 

37703 

92620 

39314 

91948 

40913 

91248 

51 

10 

34475 

93869 

36108 

93253 

37730 

92609 

39341 

91936 

40939 

91236 

50 

11 

34503 

93859 

36135 

93243 

37757  92598 

3D367 

91925 

40966 

ill  2-24 

49 

13 

34530 

93849 

36162 

9323-2 

37784 

92587 

39394 

91914 

4()<llJ-2 

91212 

48 

13 

34557 

93839 

36190 

93222 

37811 

92576 

39431 

91902 

41019 

91200 

47 

14 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

91891 

4104.') 

91188 

46 

15 

34612 

93819 

36244 

93201 

37865 

92554 

39474 

91879 

41072 

91176 

45 

16 

34639* 

93809 

36271 

93190 

37892 

92543 

39501 

91868 

41098 

91164 

44 

17 

34666 

93799 

36298 

93180 

37919 

92532 

39528 

91856 

41125 

91152 

43 

18 

34694 

93789 

36325 

93169 

37946 

92521 

39555 

91845 

41151 

91140 

42 

19 

34721 

93779 

36352 

93159 

37973 

92510 

39581 

91833 

41178 

91128 

41 

20 

34748 

93769 

36379 

93148 

37999 

92499 

39608 

91822 

41204 

91116 

40 

21 

34775 

93759 

36406 

93137 

38026 

92488 

39635 

91810 

41231 

91104 

39 

22 

34803 

93748 

36434 

93127 

38053 

92477 

39661 

91799 

41257 

91092 

38 

23 

34830 

93738 

36461 

93116 

38080 

92466 

39688 

91787 

141284 

91080 

37 

24 

34857 

93728 

36488 

931U6 

38107 

92455 

39715 

91775 

41310 

91068 

36 

25 

34884 

93718 

36515 

93095 

38134 

92444 

39741 

91764 

41337 

91056 

35 

26 

34912 

93708 

36542 

93084 

38161 

92432 

39768 

91752 

i  41363 

91044 

34 

27 

34939. 

93698 

36569 

93074 

38188 

92421 

39795 

91741 

41390 

91032 

33 

28 

34966 

93688 

38596 

93063 

38215 

92410 

39822 

91729 

41416 

91020 

i2 

29 

34993 

93677 

36623 

93052 

38241 

92399 

39848 

91718 

41443 

91008 

31 

30 

35021 

93667 

36650 

93042 

38268 

92388 

39875 

91706 

414(59 

9099(5 

30 

31 

35048 

93657 

36677 

93031 

38295 

92377 

39902 

91694 

i  41496 

90984 

29 

32 

35075  i  (J3ii47 

36704 

93020 

38322 

92366 

39928 

91683 

41522 

90972 

28 

33 

3510-2  !  93637 

36731 

93010 

38349 

92355 

39955 

91671 

41549  1  90960 

34 

35130 

9362(5 

36758 

92999 

38376 

!)-2.'!4:i 

39982 

91660 

41575 

90948 

2(5 

35 

35157 

93(516 

36785 

92988 

38103  92332  ! 

40008 

91(548 

41(10-2 

9093!) 

25 

30 

35183 

93606 

36812 

92978 

38430  92321 

40035 

91636 

41628 

90924 

24 

37 

35-211 

93596 

36839 

9-29i57 

38456 

92310 

40062 

91625 

41655 

90911 

23 

38 

35-231)  |  93585 

36867 

9295'j 

38483 

92299 

40088 

91613 

41681 

90899 

2-2 

39 

352;>(i  93575 

36894 

92945 

38510 

92287 

40115 

91601 

41707 

90P87 

21 

40 

35293 

93565 

36921 

92935 

38537 

92276 

40141 

91590 

41734 

90875 

20 

41 

35320 

93555 

36948 

92924 

38564 

92265 

40168 

91578 

41760 

90863 

19 

42 

35347 

93544 

3(5975 

82913 

38591 

92254 

40195 

91566 

i  41787 

90851 

18 

43 

35375 

93534 

37002 

9290-2 

38617  1  92243 

40221 

91555 

;  41813 

90839 

.7 

44 

35402 

93524 

37029 

92892 

38644  92-231 

40248 

91543 

141840 

90826 

16 

45 

354-29 

93514 

37056 

92881 

38671  92220 

40275 

91531 

41866 

90814 

15 

46 

35456 

93503 

37083 

92870 

38698 

92209 

40301 

91519 

41892 

90802 

14 

47 

35484 

93493 

37110 

92859 

38725  !  92198 

40328 

91508 

41919 

90790 

13 

48 

35511 

93483 

37137 

92849 

38752 

92186 

40355 

91496 

!  41945 

90778 

12 

49 

35538 

9347-2 

37164 

92838 

38778 

92175 

40381 

9  14H4 

41972 

9076(5 

11 

50 

35565 

93462 

37191 

92827 

38805 

92164 

40408 

91472 

41998 

90753 

10 

51 

35592 

93452 

37218 

92816 

38832 

92152 

40434 

91461 

42024 

90741 

9 

5-2 

35619 

93441 

37245 

92805 

38859 

92141 

40461 

91449 

42051 

90729 

8 

53 

35647 

93431 

37272 

92794 

38886 

92130 

40488 

91437 

42077 

90717 

7 

54 

35674 

934-20 

37299 

92734 

38912 

92119 

40514 

91425 

42104 

90704 

6 

55 

35701 

93410 

373-215 

92773 

38939 

92107 

40541 

91414 

42130 

90C9-2 

5 

56 

35728 

93400 

37353 

92762 

3896(5 

92096 

40567 

91402 

42156 

90680 

4 

57 

35755 

93389 

37380 

92751 

38993 

92085 

40594 

91390 

42183 

911668 

3 

5rt 

35782 

93379 

37407 

92740 

39020 

92073 

40!i21* 

91378 

142209 

901)55 

2 

59 

35810 

93368 

37434 

92723 

39046 

92062 

4064? 

913(51) 

j  42235 

90643 

1 

M 

N~CS! 

N.S. 

¥.08. 

N.S. 

N.CS. 

N.S. 

N.CS 

N.S. 

N.CS, 

~N~s7 

M 

69  Deg. 

68  Deg. 

67  Deg. 

66  Deg. 

~6lTDeg7 

NATURAL  SINES. 


25  Deg. 

26  Deg. 

27  Deg. 

28  Deg.  !  29  Deg. 

M 

N.S 

N.CS 

N.S. 

N.CS 

N.S. 

N.Cb 

N.S. 

N.CS.  i  N.S 

N.CS 

M 

0 

4226 

!K)63 

43837 

8987! 

45399 

8illO 

4694' 

8829.3  48481 

874(hT 

60 

I 

4228 

90618 

43885 

46973 

88281  4-.10( 

87448 

.19 

2 

4231J 

90601 

43881 

8985 

4.14.11 

89074 

46991 

•is;,:i-. 

87434 

58 

n 

4234 

9059 

43916 

8984 

45477 

4702- 

88254  48557 

874  2( 

57 

4 

42367 

90582 

43943 

S!)S-> 

4.1.103 

470.K 

88240  48.1K 

87406 

56 

5 

42394 

UO.ll,! 

439*68 

80811 

15521 

39035 

4707( 

8822(i  48(508 

87391 

55 

( 

(-'»-,'( 

90557 

43994 

89801 

45554 

47101 

882J3  48634 

87377 

54 

7 

4244( 

!»!W4.i 

44028 

^!>7i!; 

4.1580 

81)0!  >J- 

47127 

88199  1  48659 

87363 

53 

8 

42473 

90532 

44046 

89777 

45606 

88995 

171.1: 

88185 

48084 

87349 

52 

g 

4249! 

90.->2() 

44072 

89764 

45632 

88981 

47178 

88172 

48710 

873:55 

51 

10 

42S35 

90507 

44098 

89753 

45658 

88968 

47204 

88158 

487.11 

87321 

50 

11 

42552 

9049.1 

44124 

8973! 

4.1684 

88955 

47820 

881-14 

4H76I 

87:506 

49 

12 

42578 

9,148:5 

44151 

89726 

45710 

88942 

47255 

88130 

48786 

^7-2!  >2 

48 

13 

42604 

90470 

44177 

897J  3 

45736 

88928 

47281 

88117 

48811 

87278 

47 

14 

4263] 

90458 

44203 

89700 

45762 

88915 

47306 

88103 

48837 

87284 

46 

15 

42fi57 

90441) 

44229 

89687 

45787 

88902 

47332 

88089 

48862 

87250 

45 

16 

42683 

90433 

4-12.1.1 

89674 

45813 

88888 

47368 

980(75 

48888 

87235 

44 

17 

4271)!) 

90421 

4428] 

89662 

45839 

88875 

47383 

88062 

48913 

87221 

43 

18 

4273(5 

90408 

44307 

89649 

45865 

88862 

47409 

88048  148938 

87207 

42 

19 

42762 

9o:ii»6 

44333 

89631) 

45891 

88848 

47434 

88034  48i)i;i 

87193 

41 

90 

42788 

90:583 

44359 

89623 

45917 

S-'83.1 

7460 

8J0_>0  48989 

87178 

40 

21 

42815 

90371 

44385 

89610 

45942 

S8822 

47486 

88006  '49014 

87164 

39 

22 

42841 

J0358 

44411 

89597 

45968 

88808 

47511 

87993 

49040 

87150 

38 

2:j 

42867 

»0340 

44437 

89584 

45994 

88795 

47537 

87979 

49065 

87136 

37 

24 

42894 

JOM4  : 

44464 

89571 

46020 

-<S7H2 

7562 

87965 

49090 

87121 

36 

25 

42920 

»032I 

44490 

89558 

46046 

88768 

7588 

87951 

49116 

87107 

L5 

26 

42946 

10:509 

44516 

89545 

16072 

(8755 

7614 

87937 

49141 

87093 

34 

27 

42972 

10296 

44542 

89532 

46097 

88741 

7639 

87923 

49166 

B7OT9 

53 

28 

42999 

90284 

44568 

89519 

46123 

-•H72H  ! 

7665 

87909 

49192 

87064 

32 

29 

4302.3 

90271 

44594 

89506 

46149 

88715 

7690 

87896 

49217 

87050 

31 

30 

43051 

90259 

44620 

89493 

46175 

88701 

7716 

87882  | 

49242 

87036 

30 

31 

43077 

90246 

44646 

89480 

46201 

88688 

7741 

87868  J 

49268 

87021 

29 

32 

43104 

J0233 

44672 

89467 

6226 

88674 

7767 

87854 

49293 

87007 

28 

33 

43130 

90221 

44698 

89454 

6252 

88661 

7793 

87840 

49318 

6993 

27 

34 

43156 

10-208 

44724 

89441 

6278 

88647 

7818 

87826 

49344 

6978 

26  ' 

35 

43182 

90196 

44750 

89428 

46304 

88634 

7844 

87812 

49369 

6964 

25 

36 

43209 

J0183 

44776 

89415 

6330 

88620 

7869 

87798 

49394 

6949 

24 

37 

43235 

liHTI 

44802 

9402 

6355 

88607 

7895 

87784 

49419 

6935 

23 

38 

13261 

K)158 

1828 

6381 

88593  : 

7920 

87770 

49445 

6921 

8 

39 

J0146 

44854 

89376 

6407 

88580 

7946 

877:16 

49470 

83906 

;1 

40 

43:?]  3 

00133 

44880 

89363 

6433 

88586 

7971 

87743 

4949.1 

6802 

20 

41 

43340 

90120  ! 

44900 

89350 

6458 

88553 

7997 

87729  ;  49.121 

(1878 

9 

42 

43366 

JO  108 

44932 

89337 

6484 

88539  i 

8022 

r-771.1  •['.).>{(> 

6863 

8 

43 

43382 

)0-:>95 

4958 

89324 

li.110 

88526 

8048 

*:701  49571 

6849 

7 

44 

43418 

90082 

44984 

89311 

(5536 

H>'.ll-2 

8073 

87687  j 

4959(i 

6834 

6 

45 

43445 

90070 

45010 

WISH 

6561 

KH4W 

8099 

87673: 

49622 

86820 

5 

46 

43471 

90057 

45036 

89285 

6587 

88485 

8124 

87659 

49647 

6805 

4 

47 

43497 

90045 

.1062 

89272 

6613 

HSI72 

8150 

7645 

49672 

6791 

3 

48 

43523 

90032 

45088 

89259 

6639 

88458 

8175 

87631  j 

49697 

H777 

2 

49 

43549 

90019 

5114 

89245 

6664  !  88445 

8201 

87617  ; 

49723 

6762 

1 

50 

43575 

90007 

5140 

89232 

6690  !  88431 

8226 

87603; 

49748 

6748 

0 

51 

43602 

89994 

5166 

89219 

6716 

88417 

8252 

7389 

49773 

6733 

9 

52 

4362H 

:«HI 

5192 

89206  I 

6742 

88404 

8877 

7575 

49798 

6719 

8 

53 

43654 

9968 

5218 

89193 

6767 

HH:WO 

8303 

7561  i 

49824 

6704 

7 

54 

43680 

89956 

5243 

89180 

BT93 

88377 

8328 

7546 

49849 

6690 

6 

55 

43706 

89943 

5289 

89167 

6819 

88363 

8354 

7532 

49874 

6675 

5 

56 

43733 

89930 

5295 

89153 

6844 

88349 

8379 

87518 

49899 

6661 

4 

57 

43759 

891M8 

5321 

89140 

6870  1  88336 

8405 

7504 

49924 

6646 

3 

58 

43785 

8990.1 

5347. 

89127  1 

6896  88322 

8430 

87490 

49950 

f;t>3-2 

2 

59 

43811 

9892 

5373 

89114 

6921 

88308 

3456 

87476 

49975 

6617 

1 

M 

\.cs. 

N  S. 

N.CS. 

N.S. 

V.CS. 

N.S. 

.CS. 

N.S.  ! 

N.CS 

<.  s. 

T 

64  Deg. 

63  Deg. 

62  Deg. 

61  Deg.  | 

60  Deg. 

NATURAL    SINES. 


30  Deg. 

31  Deg. 

32  Deg. 

33  Deg. 

34  Deg. 

II 

N.S. 

N.CS 

N.S.  (N.CS 

N.S. 

N.CirJ. 

N.S. 

x.cs. 

N.8. 

N.  CS 

M 

0 

50000 

866)3 

5J504  85717 

~5-><Jy2 

848U5 

54464 

83887 

55919 

82904 

60 

1 

50025 

8(5588 

51529 

85702 

5301? 

84789 

'54488 

83651 

55:)43 

82887 

59 

2 

50050 

86573 

51554 

85(587 

53041  84774 

54513 

838M5 

55968 

8-2871 

58 

3 

50076 

83559 

51579 

85672 

53066 

84759 

54537 

83819 

55992 

8-2855 

57 

4 

50101 

8(5544 

51604 

85657 

53091 

8-1743 

54561 

a3804 

56016 

8-2KI!) 

5(5 

5 

50126 

86530 

51628 

85:542 

53115 

84728 

5458'i 

:  83788 

56040 

82822 

55 

6 

50J51 

86515 

51653 

85527 

5314G 

04712 

54610 

83772 

560(54 

fcWlKi 

54 

50176 

86501 

51678 

85612 

531(34 

84697 

54635 

83756 

56088 

82790 

53 

8 

50201 

86486 

51703 

85597 

53189 

84681 

54659 

83740 

56112 

82773 

52 

9 

50227 

86471 

51728 

855*2 

53214  \  84666 

54(583 

83724 

56136 

82757 

51 

10 

50252 

86457 

51753 

85567 

'53238  '84650 

54708 

83708 

56160 

82741 

50 

1J 

50277 

86442 

51778 

85551 

33263  184635 

54732 

83692 

56184 

82724 

49 

12 

50302 

86427 

51803 

85536 

53288 

84619 

54756 

83(576 

5(5208 

82708 

48 

13 

50327 

86413 

51828 

85521 

53312 

84604 

54781 

83660 

56233 

82ii'i-2 

47 

14 

50352 

86398 

51852 

85506 

53337 

84588 

54805 

83(545 

56256 

82675 

4(5 

15 

50377 

86384 

51877 

85491 

53361 

84573 

54829 

83(529 

56280 

82659 

45 

16 

50403 

86369 

51902 

85476 

53386 

84557 

54854 

83613 

56305 

82643 

44 

1? 

50428 

86354 

51927 

85461 

53411 

84542 

54878 

83597 

56329 

8-21)2(5 

43 

18 

50453 

86340 

51952 

85446 

53435 

84526 

54902 

83581 

56353 

82610 

42 

19 

50478 

86325 

51977 

85431 

53460 

84511 

54927 

83565 

5(5377 

82593 

41 

20 

50503 

86310 

52002 

85416 

53-184  i  84495 

549.)! 

83549 

56401 

82577 

40 

21 

50528 

86295 

52026 

85401 

.">3.w)y  84480 

54975 

83533 

56425 

82561 

3a 

22 

50553 

86281 

52051 

85385 

53534  84464 

54999 

83517 

56449 

8-2541 

38 

23 

50578 

86266 

52076 

85370 

53558  84448 

55024 

83501 

5(5473 

8-2528 

37 

24 

50603 

86251 

52101 

85355 

53583 

84433 

55048 

83485 

56497 

8251  1 

30 

25 

50628 

86237 

52126 

85340 

53607 

84417 

55072 

83469 

5(5521 

82495 

35 

26 

50654 

86222 

52151 

85325 

53f>32  ;  84402 

55097 

83453 

5(5545 

82478 

34 

27 

50679 

86207 

52175 

85310 

53656  8438S 

55121 

83437 

56569 

82462 

33 

28 

50704 

86192 

52200 

85294 

53881 

84370 

55145 

83421 

5(5593 

82446 

32 

29 

50729 

86178 

52225 

85279 

53705  84355 

551(59 

83405 

56617 

82429 

31 

30 

50754 

86163 

52250 

85264 

53730  84339 

55194 

83389 

56641 

82413 

30 

31 
32 

50779 
50804 

86148 
80133 

52275 

52299 

85249 
85234 

53754  843-24 
53779  84308 

55218 
55242 

83373 
8335(5 

56665  82396 

5(5(189  H-2380 

29 
28 

33 

50829 

861  i  9 

52324 

85218 

53804  i  84292 

56966 

83340 

56713 

82.'!(;:t 

27 

34 

50854 

86104 

52349 

85203 

53828  j  84277 

5529] 

83324 

50736 

82347 

20 

35 

50*79 

86089 

52374 

85188 

."-:K">3  84261 

55315 

83308 

56760 

i->33ii 

25 

36 

50904 

86074 

52399 

85173 

5:1*77  81245 

5533» 

832«-.» 

56784 

8-2314 

-24 

37 

509-20 

86059 

50423  ,  85157 

53902  84230 

553(53  83276 

5C808 

8-2297 

23 

38 

50954 

86045 

52448 

85142 

53928  84214 

55388  1  83260 

56832 

82281 

•22 

30 

50979 

86030 

52473 

85127 

53951  84198 

55412 

83244 

5(585(5 

8-2-204 

21 

40 

51004 

86015 

52498 

85112 

53975  84182 

55436 

83228 

56880 

82248 

20 

41 

51039 

86000 

52522 

85096 

54.000  !  84167 

55460 

83212 

56904 

82-231 

19 

42 

51054 

85985 

52547 

85081 

54024  !  84151 

55484 

83195 

5(5928 

82214 

18 

43 

51079 

85970 

52572 

85066 

54049  !  84135 

555(19 

83179 

56952 

82198 

17 

44 

51104 

85956 

52597 

85051 

54073  i  84120 

55533 

83163 

5697(5 

82181 

16 

45 

51129 

85941 

52621 

85035 

54097  i  84104 

55557 

83147 

57000 

82165 

15 

46 

51154 

85926 

52646 

85020 

54122 

84088 

55581 

83131 

57024 

82148 

14 

47 

51179 

85911 

52671 

85005 

54146 

84072 

55*505 

83115 

57047 

82132 

13 

48 

51204 

85896 

52696 

84989 

54171 

84057 

55630 

83098 

57071 

82115 

12 

49 

51229 

85881 

52720 

84974 

54195 

84041 

55654 

83082 

57095 

8-2098 

11 

50 

51254 

85866 

52745 

84959 

542-2(1 

84025 

55678 

830(5(5 

57119 

82082 

10 

51 

51279 

85851 

52770 

84943 

54244 

8400!)  j 

55702 

83050 

57143 

820(55 

9 

52 

51304 

a5836 

52794 

84928 

54269 

83994 

55726 

83034 

57  It  '-7 

82048 

8 

53 

51329 

85821 

58819 

84913 

54293 

83978 

557.50 

83017 

57191 

82032 

7 

54 

51354 

85806 

52844 

84897 

54317 

««»«•! 

55775 

83001 

57215 

8-2015 

6 

55 

51379 

85792 

52869 

84882 

54342 

83940 

55799 

82985 

57238 

81999 

5 

56 

51404 

85777 

52893 

84866 

54366 

83930 

55*23 

P29C-9 

57262 

81982 

•1 

57 

51429 

85762 

52918 

84851 

54391 

83915 

55847 

82953 

57286 

81965 

3 

58 

51454 

85747 

52  W  3 

84836 

54415 

83899 

55871. 

8-2!)3(i 

57310 

81949 

59 

51479 

85732 

58987 

84820 

54440 

83883 

55895 

8292*1 

57334 

81932 

1 

M 

S.C8. 

N.S. 

N.CS. 

N.S. 

N.CS. 

N.S.  j 

f.-CS 

N.S.  ; 

N.CS 

N.S. 

M 

59  Deg. 

58  Deg. 

57  Deg. 

56  Deg.  j 

55  Deg. 

NATURAL  SJ»E«. 


71 


35  Deg. 

36  Deg. 

37  Deg. 

38  Deg. 

!  39  Deg. 

\ 

N.S. 

N.CE 

i\  .  S. 

\.r> 

N.S. 

N.Uri 

N7sT 

N.CS.  N.S 

N.Cb 

M 

( 

573.5* 

8i9lo 

587TO 

80903 

6018! 

7986 

61561 

78801 

02!  13- 

7771. 

(50 

57381 

81891 

80885 

60205 

7984f 

6158! 

78783 

629.5" 

77(i9( 

59 

57405 

81  88-. 

58826 

8()Hii7 

60228 

79821 

61612 

787(5.5 

62971 

77678 

58 

• 

57429 

8I80J 

80851 

6025J 

7981 

01635 

78747 

(531  iOi 

77C60 

j 

574.):} 

81848 

58873 

8083: 

nii-274 

7979: 

61(558 

78729 

(i:io-2-. 

',7<;ji 

5(5 

57477 

8183; 

58896 

80811 

602« 

7:»77( 

(51081 

78711 

63015 

55 

(: 

57501 

81813 

58920 

8079; 

60321 

79758 

61704 

78694 

I>:UJ<W 

77«M 

54 

7 

57524 

817SJ8 

58:143 

80782 

80344 

79741 

(i  172(5 

78670 

(i3!)9() 

77586 

53 

g 

HITS-. 

58967 

80765 

G0367 

79723 

61749 

78(558 

63113 

775(58 

52 

!) 

57572 

81765 

58090 

80748 

60390 

7970(5 

61772 

78(540 

(13135 

77.-,.5t 

51 

10 

57591) 

81748 

5:1014 

80730 

60414 

79688 

61795 

781  i  22 

1)3  1.  -,8 

77531 

50 

Jl 

57619  H7.3J 

.59KS7 

80713 

60437 

79671 

61818 

78604 

(13  18,1 

77.-,  13 

49 

12 

57(543  *1714 

59001 

8069fl 

60460 

791553 

61841 

7858ii 

(53-J03 

77494 

4d 

13 

.57(5(57 

81698 

59084 

80(57!) 

60483 

79635 

61864 

78568 

63225 

7747(5 

J7 

14 

57691 

81681 

59108 

80ii62 

60506 

79(518 

61887 

78550 

63248 

77458 

40 

15 

57715 

BUM 

59131 

8J644 

60529 

796UO 

61909 

78532 

63271 

77439 

15 

1C, 

57738 

81647 

59154 

8.J627 

60553 

79583 

6  1932 

78514 

63293 

77421 

44 

17 

57782 

8!ii3i 

59178 

HI;  10 

60576 

79565 

61955 

78498 

r.33  ir, 

77402 

13 

Iri 

57786 

81614 

59201 

80593 

60599 

79547 

61978 

78478 

63338 

77384 

42 

19 

57810 

815OT 

59225 

80578 

W622 

79530 

6200] 

784(50 

63361 

773615 

11 

20 

57833 

81580 

59248 

805.58 

501545 

79512 

62024 

78442 

63383 

77347 

4'.) 

'21 

57857 

81503 

59272 

80541 

50(5(58 

79494 

1204(5 

78424 

(53406 

77329 

19 

22 

57881 

8154(i 

59295 

80524 

60691 

79477 

620!i9 

78105 

63428 

77310 

18 

23 

57904 

81530 

59318 

80507 

50714 

79459 

52092 

78387 

(I34.5I 

-7292 

24 

5792H 

81513 

59342 

10738 

79441 

62115 

78369 

(.3473 

77273 

;6 

23 

57952 

81496 

59365 

507*51 

79  1-J  1 

62138 

78351 

(i349fi 

-7255 

26 

57970 

81479 

S04.55 

10784 

79406 

621(50 

78333 

63518 

-7236 

4 

27 

57999 

HMii-2 

5§412 

80438 

50807 

793S8 

62183 

78315 

(53540 

77218 

3 

28 

58023 

81445 

59436 

80420 

60830 

79371 

22()r, 

78297 

63563 

-7199 

52 

29 

58047 

814-28 

59459 

80403 

HW53 

-93.53 

2329 

78279 

(53.^8.5 

77181 

j 

30 

58070 

81419 

51(182 

80386 

60876 

79335 

62251 

78261 

63608 

77162 

I) 

31 

58094 

81395 

59506 

80368 

60899 

79318 

2274 

78243 

63(530 

-7144 

29 

33 

VIM 

81378 

59529 

80351 

50922 

79300 

2297 

78225 

63053 

77125 

28 

33 

58141 

813(51 

59552 

80334 

60945 

-9282 

2320 

78206 

C,3li7.5 

-7107 

7 

34 

58165 

81344 

59576 

8031(5 

509(58 

79264 

2342 

-8188 

63698 

77088 

6 

35 

58189 

81327 

5959!) 

80299 

60991 

-9247 

2365 

78170 

63720 

"7070 

5 

36 

58-21-2 

81310 

59622 

8D-282 

51:)  1.5  ~9229 

2388 

"8152 

83742 

"7051 

4 

37 

,8j:iil 

81293 

59646 

W)2(il  , 

61038 

79211 

21  1  1 

78134 

63765 

-7033 

23 

38 

5s-2,;u 

8127(5 

59609 

80247 

51061 

79J93 

2433 

78116 

63787 

77014 

°2 

39 

5S-.K! 

812.59 

59693 

80230 

61084 

79176 

2456 

78098 

63810 

f>99(5 

1 

40 

•XM7 

8124-2 

59716 

80212 

61107 

79158 

2J79 

8(179 

63839 

-(5977 

0 

41 

58330 

81225 

59739 

8019.5 

61130 

71)140 

2.5!  12 

-8061 

63854 

C.959 

9 

42 

58354 

81208 

597(53 

80178 

1153 

79122 

2.524 

78043 

63877 

6940 

8 

43 

58378 

81191 

59786 

80160 

)1176 

7911).", 

2547 

"81)25 

6389!! 

6921 

7 

44 

58401 

81174 

59809 

81)143 

1  199 

79087 

2.570 

-8007 

(53922 

6903 

8 

45 

58425 

8J157 

59332 

80125 

61222 

79069 

2592 

-7988 

63944 

6884 

5 

46 

58449 

81140 

a9856 

80108 

61245 

79051 

2615 

~7970 

63966 

68(5(5 

4 

47 

58472 

81123 

59879 

80091 

h!268 

79033 

2638 

7958 

63989 

6847 

3 

48 

58496 

81106 

S9902 

80073 

1291 

79015 

2660 

-7934 

64011 

68-28 

0 

49 

58519 

81089 

59926 

8005*5 

1314 

78998 

2683 

77916 

64033 

68  10 

I 

50 

J8543 

81072 

>9919 

80038 

1337 

78SHU 

2706 

-7897 

64056 

(5791 

0 

51 

o8567 

81055 

.9972 

80021 

13(50 

78062 

2728 

7879 

64078 

0772 

9 

52 

58590 

81038 

9995 

80003 

1383 

7H914 

2751 

64100 

(5754 

8 

53 

8*114 

81021 

0019 

79980  ; 

1406 

78926 

HJ774 

"7843 

64123 

6735 

7 

54 

8(i37 

81001 

0042 

79988 

1  1-29 

78908 

2796 

-7824 

64145 

6717 

6 

55 

8661 

80987 

00(55 

79951 

1451 

78891 

2819 

77806 

64167 

069M 

5 

56 

08  684 

80970 

0089 

79931 

1474 

78873 

2842 

64190 

6(179 

4 

51 

o8708 

80953 

0112 

79916 

1497 

78  -Ci5 

28(54 

77789 

64212 

6601 

I 

58 

•58731 

8093(5 

0135 

79-'99 

1520 

2887 

-77.51 

64234 

6642 

o 

59 

58755 

80919 

0158 

798HI 

1543 

78819 

2909 

-7733 

6425(5 

6023 

1 

~M~ 

\.CSJ  N.S. 

N.  OS. 

N.S. 

V.CS. 

N.S. 

N.CS 

~N7s7 

N  CS.j 

N.S. 

I 

54  Deg. 

53  Deg. 

52  Deg. 

51  Deg.  , 

50  Deg. 

NATURAL  SINES. 


M 

40  Deg. 

41  Deg. 

42  Deg. 

43  Deg.    44  Deg. 

N.  S 

N.CS 

i\.S.  <N.O 

!  N.S. 

N.O 

A.  S. 

IN.  OS 

!  N.  S. 

N.CS 

M 

i 

O427'j 

70604 

65806 

754? 

I  66913 

7431 

68-20U 

73J35 

B9466 

71934  60 

- 

64301 

7i5586 

65628 

7.545- 

:  06935 

7429 

C8221 

73116 

69487 

71914  59 

t 

64323 

76567 

I  (55650 

75433 

6f!950 

7427b 

O-24*, 

73(i«J6 

!  69508 

71894  '<  58 

• 

6434f 

70548 

05672 

75414 

65978 

7425 

682G4 

73  70 

69529 

71873  !  57 

4 

64368 

76530 

65694 

7539o 

i  6  >999 

7423 

68285  7:;1)/>0 

69.349 

71853  |  56 

c 

04391) 

70511 

65716 

75375 

67021 

7421 

083(16  i  73i)3fi 

69570 

71833  55 

6 

(544J2 

76492 

65738 

7535b 

67043 

7419s 

68327 

73010 

!  09591 

71813  !  54 

64435 

76473 

65759 

75337 

67064 

74178 

68349 

729U6 

69612 

71792153 

8 

64457 

76455 

65781 

75318 

6708!) 

74159 

68370 

72976 

1  69G33 

71772  5-2  i 

•  9 

04479 

7643(5 

65803 

75291 

67107 

74139 

08391 

72;).r 

09(554 

71752 

51 

10 

04501 

76417 

65825 

75280 

67129 

74121 

;   I.*4I2 

7-2!»:r 

09075 

71732 

50 

11 

64524 

7(5398 

65847 

75201 

07151 

74h!( 

08-133 

72JH7 

09096 

71711 

49 

12 

64546 

76380 

65869 

75241 

67172 

74080 

68455 

7289- 

69717 

71091 

48 

13 

64568 

76301 

(55891 

7522-, 

67]  94 

74061 

08476 

7287" 

09737 

71671 

47 

14 

64590 

76342 

65913 

75203 

67215 

74041 

68497 

728.r 

097  5H  !  71(550 

46 

15 

64612 

76323 

65935 

75184 

67237 

74022 

08518 

72837 

i  09779 

71030 

45 

16 

64635 

76304 

65956 

75165 

67258 

7400-2 

68539 

72817 

69800 

71610 

44 

17 

64657 

70286 

(55978 

75146 

67280 

73983 

68501 

72797 

6982] 

71590 

43 

18 

01679 

70267 

6(1000 

751-26 

G7301 

73963 

68582  7-27:7 

69842 

71569 

42 

19 

64701 

76248 

O';022 

75107 

67323 

73944 

686S!3  72757 

69802 

71549 

41 

2!) 

64723 

7(5229 

66044 

75088 

67344 

73924 

T8G24  i  72737 

69883 

71529 

40 

21 

6474(5 

76210 

6i!000 

75069 

67366 

73004 

68645 

7-2717 

09904 

7I50H 

39 

22 

64708 

76192 

06088 

75050 

07387 

73885 

68000 

72i5l»7 

69925 

71488 

38 

23 

64790 

76173 

66109 

75030 

67409 

7:?865 

68088  7-07' 

09940 

71468 

37 

24 

64812 

76154 

66131 

75011 

67430 

73340 

68709  72057 

09900 

71447 

3(5 

25 

64834 

76135 

68153 

74992 

67453 

73820 

08730  7-2637 

09987 

7  J  427 

35 

26 

04850 

76116 

66175 

74973 

67473 

73800 

68751 

72(517 

70008 

71-407 

34 

27 

6-1878 

761)97 

G6197 

74953 

67495 

73787 

C8772  I  72597 

70029 

7138(5 

33 

28 

64901 

76078 

66218 

74934 

67516 

73767 

68793  i  72577 

70049 

71366 

32 

20 

54923 

76059 

66240 

74915 

67538 

73747 

688  L4 

72557 

70070 

71345 

31 

30 

61945 

70041 

66202 

74893 

67559 

73728 

68835 

72537 

7009  L 

71325 

30 

31 

64967 

76022 

66284 

74876 

67580 

73708 

68857 

72517 

70112 

71305 

29 

32 

54989 

76003 

66306 

74857 

(57002 

73088 

68878 

72497 

70132 

71084 

28 

33 

J5011 

75984 

00327 

74838 

67623 

7:;;;r.;> 

68899  I  72477 

70153  71204 

27 

34 

05033 

75965 

06349 

74818 

67645 

73049 

<>•!  1-20  72457 

70174 

71243 

20 

35 

55055 

~-M<\'  60371 

74799 

67000 

73029 

68941  i  72437 

70195 

71-223 

25 

36 

55077 

75927  !i  66393 

74780 

67688 

73010 

68902  72417 

70215 

71203 

24 

37 

65099 

75908 

66414  !  74700 

67709 

73590 

68983  1  72397 

70236 

71182 

23 

3S 

15122 

75889 

6043:5 

74741 

67730 

73570 

09004  I  72377 

70257 

71102 

22 

39 

65144 

75870 

(56458 

74722 

67752 

73551 

091)25  72357 

70277 

71141 

21 

40 

65166 

75851 

6648) 

74703 

07773 

73531 

(HW4<5  72337 

70298 

71121 

20 

41 

65188 

75832 

6(5501 

74083 

67795 

73511 

69067  72317 

70319 

71100 

19 

42 

05210 

75813 

6(55-23 

74!504 

17816 

73491 

69088  72297 

70339 

71080 

18 

43 

65232 

75794 

6r.545 

74(544 

67837 

73472 

691Q9 

72277 

703(50 

71059 

17 

44 

55254 

~577.1.  60566 

74025 

67859 

73452 

69130 

72257 

70381 

7JOH9 

16 

45 

65276 

75756  66588 

74606 

67880 

73432 

69151 

7223(5 

70401 

71019 

15 

4f> 

65238 

75738  ||  66610 

74580 

67901 

73412 

69172 

72216 

70422 

70998 

14 

47 

65320 

"'•719  66(532 

74507 

67923 

73393 

69193  72196 

70443 

70978 

13 

48 

05342 

-5699 

6S653 

74548 

57944 

73373 

69214 

72176 

70463 

70957 

12 

49 

05364 

75080  ; 

66675 

74528 

67905 

73353 

69235 

72156 

70484 

70937 

11 

50 

55381! 

7o-;>:;i 

66697 

74509 

67987 

73333 

69256 

72136 

"0505 

7091(5 

10 

51 

55408 

7.V54-2 

66718  74489 

08008 

73314 

69277 

72116 

-0525 

70896 

9 

52 

55430 

7.-.'  123 

6  5740 

74470 

680-29 

73294 

59298 

72095 

70546 

70875 

8 

53 

65452 

75004 

66702 

74451 

08051 

73274 

09319 

72075 

-0567 

70855 

7 

54 

65474 

75585  ; 

60783 

74431 

58072 

73254 

69340 

72055 

"0587 

70834 

6 

55 

35496 

75566  !  66805 

74412 

58093 

73234 

69301 

72035 

-0608 

7(1813 

5 

56 

05518 

75547:  66827 

74392 

68115 

73215 

19332 

72015 

70028 

70793 

4 

57 

05540 

75528'  06848 

74373 

68130 

73195 

69403 

71995 

-0649 

70772 

3 

58 

6r>562 

75509  !  61870 

74353 

68157 

73175 

(9484 

71974 

-0670 

70752 

2 

59 

65584 

75190  60891 

74334 

58179 

73155 

59445 

71954 

-0090 

70731 

1 

60 

5560S 

75471  1.  66913 

74314 

68200 

73135 

094(50  71934 

70711 

70711 

IT 

v.cs. 

N.S.  j:\.CS. 

N.S. 

N.CS. 

N.S. 

\T.  OS  |  N.  S. 

N.OS.I  N.S. 

V! 

49  Deg.  Jl  48  Deg. 

47  Deg. 

46  Deg. 

45  Deg. 

NATCRAL  TAN'GKNTS. 


0  Degrees. 

1  Degree. 

2  Degrees. 

3  Degrees. 

M 

N.Tan. 

N.  Cot. 

X.  IVi. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan 

N.  Cot. 

M 

0 

00000 

oooo.oo 

~oT74tT 

57.2900 

03492' 

28.63  i.t 

05241 

79.0811 

"W 

1 

00.12:1 

3437.75 

1)177.1 

56.3506 

0352  1 

28.3991 

05270  ' 

18.9755 

59 

o 

00058 

1718.87 

01804 

55.4415 

03550 

28.16(54 

05299 

18.8711 

58 

3 

00067 

ii4.-i.y-j 

01833 

54.5613 

03579 

27.9372 

053-28 

18.7678 

57 

4 

00116 

859.436 

OM62 

53.7086 

03609 

27.7117 

05357 

18.6656 

56 

5 

00115 

687.54g 

01891 

03638 

•27.48'MI 

05387 

18.5645 

55 

6 

OD  IT.-) 

572.957 

019-20 

52.0801 

03667 

27.2715 

05416 

18.4(545 

54 

00204 

491.  106 

01949 

51.303-2 

036!)!i 

27.0566 

05445 

18.  365.") 

53 

8 

00233 

429.718 

01978 

50.5485 

03725 

26.8450 

05474 

18.2677 

52 

9 

00202 

381.971 

0-2007 

49.  HI  57 

03754 

26.6367 

05503 

18.1708 

51 

10 

00291 

343.774 

02036 

49.  1039 

03783 

2U.4316 

05533 

18.0750 

5.1 

11 

0032  ) 

312.521 

02066 

48.41-21 

03812 

20.229G 

05562 

L7.98Q] 

49 

1-2 

80349 

02095 

47.7395 

03842 

2(5.0307 

05591 

17.8863 

48 

13 

00378 

264.441 

02124 

47.0853 

03871 

25.8348 

05620 

17.7934 

47 

J4 

90407 

24.->.  55-2 

02153 

46.  448  J 

03900 

25.6418 

05649 

17.7015 

46 

15 

004313 

229.182 

02182 

45.8294 

03929 

25.4517 

05678 

17.6106 

45 

10 

00465 

214.858 

0-2211 

45.2261 

03958 

25.2644 

05708 

17.5205 

44 

17 

00495 

202.  -2  lit 

(1-224!) 

•4.  038!) 

03947 

25.0798 

05737 

17.4314 

43 

18 

00524 

191).  984 

02269 

44.0661 

04016 

24.89-78 

05766 

17.3432 

42 

19 

DO.-).-):? 

180.932 

02298 

43.5081 

0404(i 

24.7185 

05795 

17.2558 

41 

JO 

00582 

171.885 

02328 

42.91)41 

04075 

24.5418 

05824 

17.1693 

40 

21 

O.Kill 

163.700 

02357 

42.4335 

04104 

24.3675 

05854 

17.0837 

39 

22 

00640 

150.251) 

02386 

41.9158 

04133 

24.1957 

05883 

16.9990 

38 

23 

001)61) 

149.  465 

0-2415 

41.4106 

04162 

•24.0-2ii:i 

05912 

]  (5.  9  150 

37 

24 

00698 

14  3.  -2:57 

02444 

•1  1.9174 

04191 

23.8593 

05941 

16.8319 

36 

2.) 

D07-27 

137.507 

0-2473 

40.4358 

042-20 

23.6945 

05970 

16.7496 

35 

2li 

007515 

13-2.219 

02503 

39.9655 

04250 

•2::.5:i-2i 

05999 

16.6681 

34 

27 

1-27.3-21 

02531 

3;  1.505!) 

04279 

23.3718 

06029 

16.5874 

33 

28 

00814 

1-2-2.774 

02530 

39.0568 

04308 

23.2137 

06058 

16.5075 

32 

29 

0084  i 

118.540 

02589 

38.6177 

04337 

23.0577 

06087 

16.4283 

31 

30 

00873 

114.589 

02619 

38.1885 

04366 

22.9037 

06116 

16.3499 

30 

31 

00903 

110.892 

02648 

37.7686 

04395 

22.7518 

06145 

16.2722 

29 

32 

00931 

107.420 

02677 

37.3579 

04424 

22.6020 

06175 

16.1952 

28 

33 

00960 

104.171 

02706 

36.95(iO 

04454 

22.  4541 

06204 

16.1190 

27 

31 

OOU89 

101.107 

0-27.!.") 

36.5627 

04483 

22.3081 

06233 

16.0435 

26 

35 

01018 

98.2179 

0-2764 

36.1776 

04512 

22.1640 

06262 

15.9687 

25 

36 

01047 

95.4895 

0-2793 

35.8006 

04541 

2-2.0-217 

08291 

15.8345 

24 

37 

01076 

92.0085 

02822 

35.4313 

04570 

21.8813 

06321 

15.  8-2  II 

23 

38 

01105 

!)!».  41533 

02851 

35.0695 

04599 

21.7426 

08350 

15.7483 

22 

39 

01135 

88.1436 

02831 

34.7151 

04(528 

21.6056 

015379 

15.  (57(5-2 

21 

;  i 

01164 

B5u939S 

0-21(10 

34.3678 

04658 

21.4704 

06408 

[5.6948 

20 

41 

01193 

83.8435 

02939 

34.0-273 

04687 

21.3369 

00437 

15.5340 

19 

42 

0  1-22-2 

81.8470 

02968 

33.6935 

04716 

21.2049 

0(5467 

15.4638 

18 

43 

01251 

79.9434 

02997 

33.3662 

04745 

21.0747 

06496 

15.3943 

17 

44 

01283 

7*.  1263 

03026 

33.0452 

01774 

20.9460 

06525 

15.3954 

16 

45 

01309 

76.3900 

03055 

32.7303 

04803 

20.8188 

06554 

15.2571 

15 

40 

01338 

74.7292 

03084 

32.4213 

04832 

20.6932 

06584 

15.1893 

14 

47 

01367 

73.  13  JO 

03114 

:!-2.M8i 

04862 

20.5691 

06613 

L5.1222 

13 

48 

0139(i 

71.6151 

03143 

31.8205 

04891 

20.4465 

0(5642 

15.0557 

12 

49 

01425 

70.1533 

03172 

31.5284 

04920 

20.3253 

06671 

14.9898 

11 

51) 

01455 

68..  75:  >  I 

03201 

31.  -24  l(i 

04949 

20.205;: 

06700 

L4.9244 

10 

51 

01484 

67.4019 

03230 

30.9599 

04978 

20.087-2 

0(5730 

14.8596 

9 

52 

01513 

66.1055 

03259 

30.6833 

05007 

19.9702 

06759 

14.7954 

8 

53 

01542 

64.8580 

03288 

30.4116 

05037 

06788 

14.7317 

7 

54 

01571 

63.6567 

03317 

30.1446 

05066 

19.7403 

0(5817 

14.6685 

6 

55 

01600 

62.4992 

03346 

29.8823 

05095 

19.6273 

06847 

14.6059 

5 

515 

01629 

6l.38/2!t 

03376 

29.6215 

05124 

I9;5156 

06876 

L4.5438 

4 

57 

(i:).:!<).->8 

03405 

29.3711 

05153 

19.40)1 

06905 

1  1.4^23 

3 

58 

01687 

59.  -2<  ;.-,•) 

n3i3» 

•2.(.!'J2!) 

05H-2   '    19.2959 

06934 

14.4218 

2 

59 

01716 

58.2312 

o:!4(>:? 

0.7212 

n.H:;) 

05983 

L4.3807 

1 

60 

01746 

57.2900 

2-UKJt;3 

05241 

19.0811 

06993 

U.3007 

0 

M 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  T;iii.; 

N.  Cot. 

N.  T.tu. 

N.  Cot. 

N.  Tan. 

M 

89  Degrees. 

88  Degrees. 

87  Degrees. 

86  Degrees. 

NATURAL,  TANGENTS. 


4  Degrees. 

5  Degrees. 

6  Degrees,     if     7  Degrees. 

M 

N.Tan 

N.  Cot 

N.Tan 

N.  Cot. 

N.Tai 

N.  Cot.     N.Tan 

N.Cot.     M 

0 

OU993 

14.300 

087-il) 

11.430 

1051U 

9.51430 

1-2273 

8.14435    tiO 

1 

07022 

14.241 

08778 

11.3919 

10540 

9.48781 

12308 

8.1248J1   59 

2 

07051 

14.18-2 

08807 

11.3541 

10569 

9.46141 

12338 

8.1053( 

;  58 

3 

07080 

14.123 

08837 

11.3163 

10599 

9.43515 

12367 

8.0860( 

)    57 

4 

07110 

14.065 

08866 

11.2789 

106-28 

9.40904 

12397 

8.0667- 

I    56 

5 

07139 

14.007 

08895 

11.241- 

10657 

9.38307 

12426 

8  .  0475( 

55 

6 

07168 

13.950 

089-25 

11.2048 

10687 

9.3572 

12456 

8.0284? 

54 

7 

07197 

13.894 

08954 

11.1681 

10716 

9.3315 

12485 

8.0094* 

53 

8 

07227 

13.837 

08983 

11.1316 

10746 

9.3059 

12515 

7  .  99056 

5-2 

0 

07256 

13.782 

09013 

11.0954 

10775 

9.2805 

12544 

7.97176    51 

10 

07285 

13.726 

09042 

11.0594 

10805 

9.2553 

12574 

7.95302    50 

11 

07314 

13.671 

09071 

11.0237 

10834 

9.2301 

12603 

7.934381  49 

12 

07344 

13.617 

09J01 

10.9881 

10863 

9.2056 

12633 

7.9158x 

48 

13 

07373 

13.563 

09130 

10.9528 

10893 

9.1802 

12662 

7.89734 

47 

14 

07402 

13.5098 

09159 

10.9178 

10922 

9.15554 

12692 

7.87895 

46 

15 

07431 

I3.456b 

09189 

10.8829 

10952 

9.1309 

12722 

7.80064 

45 

16 

07461 

13.4039 

09218 

10.8483 

10981 

9.1064 

12751 

7.84242 

44 

17 

07490 

13.3515 

09247 

10.8139 

11011 

9.0821 

12781 

7.82428 

43 

18 

07519 

13.2996 

09277 

10.7797 

11040 

9.05789 

12810 

7.80622 

42 

19 

07548 

13.2480 

09306 

10.7457 

11070 

9.03379 

12840 

7.78825 

41 

20 

07578 

13.1969 

09335 

10.71J9 

11099 

9.0098: 

12869 

7.77035 

40 

21 

07607 

13.1461 

09365 

10.6783 

11128 

8.98598 

12893 

7.75254 

39 

22 

67636 

13.0958 

09394 

10.6450 

11158 

8.96227 

129-29 

7.73480 

38 

23 

07665 

13.0458 

09423 

10.6118 

11187 

8.9386" 

12958 

7.71715 

37 

24 

07695 

12.9962 

09453 

10.5789 

11217 

8.91520 

12988 

7.69957 

36 

25 

07724 

12.9469 

09482 

10.5462 

11246 

8.89185 

13017 

7.68208 

35 

26 

07753 

12.8981 

09511 

10.5136 

11276 

8.86862 

13047 

7.66466 

34 

27 

07782 

12.8496 

09541 

10.4813 

11305 

8.84551 

13076 

7.64732 

33 

28 

07812 

12.8014 

G9570 

10.4491 

11335 

8.82252 

13106 

7.63005 

32 

29' 

07841 

12.7536 

09600 

10.4172 

11364 

8.79964 

13136 

7.61287 

31 

30 

07870 

12.7062 

09629 

10.3854 

11394 

8.77689 

13165 

7.59575 

30 

31 

07899 

12.6591 

09058 

10.3538 

11423 

8.75425 

13195 

7.57872 

29 

32 

07929 

12-6124 

09688 

10.3224 

11452 

8.73172 

13224 

7.56176 

28 

33 

07958 

12.5660 

09717 

10.2913 

U  482 

8.70931 

13254 

7.54487 

27 

34 

07987 

12.5199 

09746 

10.2602 

H511 

8-68701 

13284 

7.52806 

26 

35 

08017 

12.4742 

09776 

10.2294 

U541 

8.66489 

13313 

7.51132 

25 

36 

08046 

12.4288 

09805 

10.1988 

H570 

8.64275 

13343 

7.49465 

24 

37 

08075 

12.3838 

09834 

10.1683 

11600 

8.02078 

13372 

7.4780fr 

23 

38 

08104 

12.3390 

09864 

10.1381 

U629 

8.59893 

13402 

7.46154 

22 

3'J 

08134 

12.2946 

09893 

10.1080 

11659 

8.57718 

13432 

7.44509 

21 

40 

08163 

12.2505 

09923 

10.0780 

11688 

8.55555 

13461 

7.42871 

20 

41 

08192 

12.2067 

09952 

10.0483 

11718 

8.53402 

13491 

7.41240 

19 

42 

08221 

12.1632 

09981 

10.0187 

11747 

8.51259 

13521 

7.39616 

18 

43 

0825^1 

12.1201 

10011 

9.98930 

11777 

8.49128 

13550 

7.37999 

17 

44 

08280 

12.0772 

10040 

9.96007 

11806 

8.47007 

13580 

7.36389 

16 

45 

08309 

12.0346 

10069 

9.93101 

11836 

8.44896 

13609 

7.34786 

15 

46 

08339 

11.9923 

10099 

9.90211 

11865 

.".4-2795 

13639 

7.33190 

14 

47 

08368 

11.9504 

10128 

9.87338 

11895 

8.40705 

13669 

7.31600 

13 

48 

08397 

11.9087 

10158 

9.84482 

11924 

8.38625 

13698 

7.300J8 

12 

49 

08427 

11.8673 

10187 

9.81641 

11954 

8.36555 

13728 

7.28442 

11 

50 

08456 

11.8262 

10216 

9.78817 

11983 

8.  34496  ! 

13758 

7.26873 

10 

51 

08485 

11.7853 

10246 

9.76009 

12013 

8.32446 

13787 

7.25310 

9 

52 

03514 

11.7448 

10275 

9.73217 

12042 

8.30406 

13817 

7.23754 

8 

53 

08544 

11.7045 

10305 

9.70441! 

13073 

8.28376 

13846 

7.22204 

7 

54 

08573 

11.6645! 

10334 

9.  67680  i 

12101 

8.21)355 

13876 

7-20061!     f, 

55 

08602 

11.6248 

10363 

9.64935! 

12131 

8.34345 

13906 

7.191251     5 

56 

08632 

11.5853 

10393 

9.62205! 

121(30 

8.22344 

13935 

7.17594 

4 

57 

08661 

11.5461 

10422 

9.  59490  ' 

12190 

8.20352 

13965 

7.16071 

3 

58 

08690 

11.5072 

10452 

9.  56791  i 

12219 

8.18370 

13995 

7.14553 

2 

59 

08720 

11.4685 

10481 

9.54106 

13349 

8.16398 

14024 

7.13042 

1 

60 

08749 

11.4301 

10510 

9.51436 

12278 

8.14435 

14054 

7.11537 

0 

M 

N  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan. 

N.  Cotj  N.  Tan. 

N.  Cot. 

N.  Tan. 

M 

85  Degrees. 

84  Degrees.    , 

83  Degrees. 

82  Degrees. 

NATURAL  TANGENTS. 


75 


8  Degrees. 

9  Degrees. 

10  Degrees. 

11  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan 

N.  Cot. 

M 

0 

14054 

7.11537 

15838 

6.31375 

17633 

5.6712* 

19438 

5.14455 

(50 

1 

14084 

7.1003H 

15888 

6.30189 

17663 

5.  tit  5  1(55 

19486 

5.13658 

59 

2 

14113 

7.08546 

15898 

6.29007 

17693 

5.65205 

19498 

5.12862 

58 

3 

14143 

7.07059 

15928 

6.27H29 

17723 

5.64248 

19529 

5.12069 

57 

4 

14173 

7.05579 

15958 

15.26655 

17753 

5.63-295 

19559 

5.11279 

56 

5 

14202 

7.04105 

15988 

6.  25486 

17783 

5.62344 

19589 

5.10490 

55 

G 

142*2 

7.K2037 

16017 

6.24321 

17813 

5.61397 

19619 

5.0!)704 

54 

1 

14262 

7-01174 

16047 

6.  23  160 

17843 

5.60452 

19649 

5.08921 

53 

8 

14291 

6-99718 

16077 

6.2200:5 

17873 

5.59511 

19680 

5.08139 

52 

9 

j  43-2  1 

6-982(58 

16107 

6.20851 

17903 

5.58573 

19710 

5.073(50 

51 

10 

14351 

6-968-23 

16137 

6.19703 

17933 

5.57638 

19740 

5.06584 

50 

11 

14381 

6.95385 

16167 

6.18559 

17963 

5-56706 

19770 

5.05809 

49 

12 

14410 

6-9395-2 

16196 

6.17419 

17993 

5-55777 

19801 

5.05037 

48 

13 

14440 

H.92.V25 

162-26 

6.16283 

18023 

5.54851 

19831 

5.04267 

47 

14 

14470 

6.91104 

1Q356 

6.15151 

18053 

5-53927 

19861 

5.03499 

46 

15 

14499 

6-89688 

16286 

6.14023 

18083 

5-53007 

19891 

5.02734 

45 

16 

14529 

6.88278 

16316 

6.12899 

18113 

5.52090 

19921 

5.01971 

44 

17 

14559 

6-  86874 

16346 

6.11779 

18143 

5.51176 

19952 

5.01210 

43 

18 

14588 

6-85475 

16376 

6.10664 

18173 

5.50264 

19982 

5.00451 

42 

19 

14618 

6.84082 

16405 

6.09552 

18203 

5.49356 

20012 

4.99695 

41 

20 

14648 

6.82694 

16435 

6.08444 

18233 

5.48451 

20042 

4.98940 

40 

21 

14678 

6-81312 

16465 

6.07340 

18263 

5.47548 

20073 

4.98188 

39 

22 

14707 

6.79936 

16495 

6.06240 

18293 

5.46648 

20103 

4.97438 

38 

23 

14737 

6-78564 

16525 

6.05143 

18323 

5.45751 

20133 

4.96690 

37 

24 

14767 

6.77199 

16555 

6.04051 

18353 

5.44857 

20164 

4.95945 

36 

25 

14796 

6.75838 

16585 

6.0-«I62 

18383 

5.43966 

20194 

4.95201 

35 

26 

14826 

6.74483 

16615 

6.01878 

18414 

5.43077 

20224 

4.94460 

34 

27 

14856 

6.73133 

16645 

6.00797 

18444 

5.42192 

20254 

4.93721 

33 

28 

14886 

6.71789 

16674 

5.99720 

18474 

5.41309 

20285 

4.92984 

32 

29 

14915 

6.70450 

16704 

5.93646 

18504 

5.40429 

20315 

4.92249 

31 

30 

14945 

6.69116 

16734 

5.97576 

18534 

5.39552 

20345 

4.91516 

30 

31 

14975 

6.67787 

16764 

5.96510 

18564 

5.38677 

20376 

4.90785 

29 

32 

151)05 

6.«6463 

16794 

5.95448 

18594 

5.37805 

20406 

4.90056 

28 

33 

15034 

6.65144 

16824 

5.  943  JO 

18624 

5.36936 

2J436 

4.89330 

27 

34 

15064 

6.63831 

16854 

5.93335 

18654 

5.36070 

20466 

4.88605 

26 

35 

15094 

6.152523 

16884 

5.99383 

18684 

5.3520(5 

20497 

4.87882 

25 

36 

15124 

6.61219 

16914 

5.91235 

18714 

5.34345 

20527 

4.87162 

24 

37 

15153 

(>.5<M»-2I 

115914 

5.91)191 

18745 

5.33487 

20557 

4.86444 

23 

38 

15183 

6.58627 

16974 

5.89151 

18775 

5.32631 

20588 

4.R5727 

22 

39 

15-213 

8.57339 

17004 

5.88114 

18805 

5.31778 

20618 

4.85013 

21 

40 

15-243 

6.5(5055 

17033 

5.87080 

18835 

5.30928 

20648 

4.84300 

20 

41 

15-272 

6.54777 

17063 

5.86051 

18865 

5.30080 

20679 

4.83590 

19 

42 

15302 

6.53503 

17093 

5.85024 

18895 

5.29235 

20709 

4.82882 

18 

43 

153:5-2 

6.52-234 

17123 

5.84001 

18925 

5.28393 

20739 

4.82175 

17 

44 

15362 

6.50970 

17153 

5.82982 

18955 

5.27553 

20770 

4.81471 

16 

45- 

15391 

6.49710 

17183 

5.81966 

18986 

5.26715 

20800 

4.80769 

15 

46 

15421 

6.48456 

17213 

5.80953 

19016 

5.25880 

20830 

4.80068 

14 

47 

15451 

6.47206 

17-243 

5.79944 

19046 

5.25048 

20861 

4.79370 

13 

48 

151S1 

6.45961 

17-273 

5.78938 

19076 

5.24218 

20891 

4.78673 

12 

49 

15511 

6.44720 

17303 

5.77936 

19106 

5.23391 

20921 

4.77978 

11 

50 

15540 

6.43484 

17333 

5.76937 

19136 

5.  22566 

20952 

4.77286 

10 

51 

15570 

6.4-2-253 

17363 

5.75941 

19166 

5.21744 

20982 

4.76595 

9 

52 

15600 

6.410-26 

17393 

5.74949 

19197 

5.20925 

21013 

4.75906 

8 

53 

15630 

6.39804 

17423 

5.73960 

19227 

5.20107 

21043 

4.75219 

7 

54 

15  ;i;i) 

6.38587 

17453 

5.72974 

19257 

5.19293 

21073 

4.74534 

6 

55 

15!>89 

6.37374 

17483 

5.71992 

19-287 

5.18480 

21104 

4.73851 

5 

5(5 

15719 

6.36165 

17513 

5.71013 

19317 

5.17671 

21134 

4.73170 

4 

57 

15749 

<>.:M!t;i 

17543 

5.70037 

19347 

5.16863 

21164 

4.72490 

3 

58 

15779 

6.33761 

17573 

5.69064 

19378 

5.10058 

21195 

4.71813 

2 

59 

1580!) 

6.3-2566 

17603 

5.68094 

19408 

5.15256 

21225 

4.71137 

1 

(50 

15833 

6.31375 

17633 

5.671-28 

19438 

5.14455 

21256 

4.70463 

0 

M 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.Tan. 

N.  Cot.1  N.  Tan. 

N.  Cot.|  N.  Tan. 

~M~ 

81  Degrees. 

80  Degrees. 

79  Degrees. 

78  Degrees. 

NATURAL  TANGENTS. 


12  Degrees. 

13  Degrees. 

14  Degrees.    II    15  Degrees.    > 

M 

N.Tan. 

N.  Cot. 

N.Tan 

N.  Cot. 

N.Tai 

N.  Cot.     \.T;m 

A  .  Cot.  i    M 

0 

21250 

4.7040: 

23087 

4.33148 

249:  S3 

4.01078 

20795 

3.73205    60 

1 

4.r<t7i>i 

23117 

4.32573 

24964 

4.00582 

26826 

3.12771 

59 

o 

213  1(3 

4.0;»12I 

23148 

4.  320U  1 

24995 

4.(i0086 

26857 

3.7S338 

58 

3 

21347 

4.084.V- 

23179 

4.31430 

25020 

3.99592 

26888 

3.7190- 

57 

4 

•21377 

4.67781 

23209 

4.3080* 

25056 

3.99099 

26920 

3.7147( 

50 

5 

21408 

4.071-21 

23240 

4.:>02'Ji 

25087 

3.98007 

20951 

3.71(M( 

55 

0 

2J438 

4.00458 

23271 

4.29724 

25118 

3.98117 

2i;;is2 

3.700H 

54 

7 

214C9 

4.  05791 

23301 

4.21M5S 

251  J9 

3.97627 

27013 

3.70'188 

53 

8 

21499 

4.05138 

23332 

4.28595 

25180 

3.97139 

27044 

3.097GJ 

52 

9 

21.329 

4.154481 

23303 

4.2803- 

25211 

•3.  <)f.<v>  1 

27070 

3.69335 

51 

10 

21500 

4.63825 

23393 

4.27471 

252-12 

3.96165 

27107 

3,6890! 

50 

]JL 

21590 

4.0317 

23424 

4.21)911 

25273 

3.95680 

27138 

3.08-18.1 

4!t 

12 

21(521 

4.02518 

23455 

4.20352 

25304 

3.95196 

27109 

3.08001 

48 

13 

2l(i51 

4.61868 

23484 

4.25795 

25335 

3.94713 

27201 

fctJTflsa 

47 

14 

2ir,s-_> 

4.01-211 

23516 

4.252::9 

25300 

3.94232 

27232 

3.072)7 

40 

15 

21712 

4.00572 

23547 

4.24085 

25397 

3.93751 

27203 

3.66796 

45 

10 

21743 

4.59927 

23578 

4.24132 

25428 

3.93271 

2721)4 

3.66376 

44 

17 

21773 

4.59283 

23008 

4.23580 

25459 

3.92793 

27320 

3-65957 

43 

18 

21804 

4.58041 

23039 

4.23030 

25490 

3.92310 

27357 

3.  05538 

42 

19 

21834 

4.58001 

23070 

4.22-181 

25521 

3.91839 

27388 

3.05121 

41 

20 

21804 

4.57303 

23700 

4.21933 

25552 

3.91304 

27419 

3.64705 

40 

21 

21895 

4.5(57-21 

23731 

4.21387 

25583 

3.90890 

27451 

3.64289 

:-;9 

22 

21925 

4.50091 

231762 

4.20842 

25014 

3.90417 

27482 

3.03874 

38 

23 

21956 

4.55458 

23793 

4.20298 

25645 

3.89945 

27513 

3.0^101 

37 

24 

21980 

4.5482J 

23823 

4.19750 

25676 

3.89474 

27545 

3.63048 

30 

25 

22017 

4.54191 

23854 

4.19215 

25707 

3.89004 

27570 

3.02030 

35 

20 

22047 

4.53508 

23885 

4.18675 

25738 

3.88536 

27607 

3.02224 

34 

27 

22078 

4.52941 

23916 

4.18137 

25709 

3.88008 

27038 

3.01814 

33 

2-8 

2=2108 

4.523K 

23946 

4.17000 

25800 

3.87601 

27(570 

3.01405 

32 

29 

'22139 

4.51093 

23977 

4-17004 

25831 

3.87136 

27701 

3.00990 

31 

30 

2-2169 

4.51071 

24008 

4.1  0530 

25862 

3.86671 

27732 

3.60588 

30 

31 

222GO 

4.50451 

24039 

4.15997 

25893 

3.86208 

27764 

3.00181 

29 

32 

22231 

4.J19832 

24009 

4.15J05 

25924 

3.85745 

27795 

3.59775 

28 

33 

22261 

4.49215 

24100 

4.14934 

25955 

3.85283 

27826 

3.59370 

27 

34 

22392 

4.48DOO 

24131 

4.14405 

25986 

3.84824 

27858 

3.58966 

26 

35 

2-2:5-22 

4.47980 

2410-2 

4.13877 

26017 

3.84364 

27889 

3.58562 

25 

36 

22353 

4.47374 

24193 

4.13350 

26048 

3-83904 

27920 

3.58160 

24 

37 

22383 

4.  407154 

24223 

4.12825 

20079 

3-83449 

27952 

3.57758 

23 

38 

22414 

4.40155 

24254 

4.1230] 

26110 

3-82992 

27983 

3.57357 

22 

39 

22444 

4.45548 

24285 

4.11778 

20141 

3.82537 

28015 

3.50957 

21 

40 

22475 

4.44942 

24316 

4.11250 

20172 

3-82083 

28046 

3.50557 

20 

41 

22505 

4.443:i8 

24347 

4.10736 

20203 

3-8J630 

28077 

3.56159 

19 

42 

22530 

4.43735 

24377 

4.10216 

20235 

3-81177 

28109 

3.55761 

18 

43 

22507 

4.43134 

24408 

4.09099 

20200 

3.80726 

28140 

3.55364 

17 

44 

2-2597 

4.42534 

24439 

4.09182 

20297 

3-80276 

28172 

3.54968    16 

45 

22628 

4.41936 

24470 

4.08006 

26328 

3-79827 

28203 

3.54573J   15 

4G 

2-2658 

4.41340 

24501 

4.08152 

20359 

3.79378 

28234 

3.54179    14 

47 

22689 

4.40745 

24532 

4.07639 

20390 

3.78931 

28266 

3.53785    13 

48 

227  J  9 

4.40152 

24502 

4.07127 

2(5421 

3.78485 

28297 

3.53393    12 

49 

22750 

4.3«5t;o 

24593 

4.06616 

20452 

3.78040! 

28329 

3.53001    11 

50 

22781 

4-389H9 

24024 

4.06107 

20483 

3.77595 

28360 

3.52609'   10 

51 

22811 

4.38381 

24055 

4.05599 

20515 

3.77152 

28391 

3.52219!     9 

52 

22842 

4.37793 

24086 

4.05092; 

20546 

3.70709! 

28423 

3.51829      8 

53 

22872 

4.37207 

24717 

4.04.VI5 

20577 

3.70268 

28454 

3.51441      7 

5^ 

22903 

4.36023 

24747 

4.04081 

26008 

3.75828 

28486 

3.51053      0 

55 

2-2934 

4.36040 

24778 

4.03578' 

20639 

3:75388 

28517 

3..W06      5 

5« 

22954 

4.35459 

24809 

4.03075! 

26670 

3.  74950  1 

28549 

3.50279      4 

57 

2-2995 

4.34879 

24840 

4.02574 

20701 

3.74512 

28580 

3.49894      3 

58 

23026 

4.343'M> 

24871 

4.02074 

26733 

3.74075; 

28012 

3.49509      2 

59 

23056 

4.33723 

24902 

4.01570! 

26764 

3.73644; 

28043 

3.49125      1 

GO 

23087 

4.33148 

24933 

4.01078 

26795 

3.73205 

28675 

3.48741      0 

M 

\.  Cot. 

N.  Tan 

N.  Cot. 

N.Tan.' 

N.  Cot. 

N.  T;tn. 

N   Cot. 

N.  Tem. 

M 

77  Degrees. 

76  Degrees. 

75  Degrees. 

74  Degrees. 

NATURAL    TANGENTS. 


77 


M 

16  Degrees. 

17  Degrees. 

18  Degrees. 

19  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.  Ta  i. 

i\.  Cot. 

N.  T,m. 

iM.Cot. 

N.Tan 

N.  Ool. 

~o~ 

281)75 

3.48741 

30.>73 

3.27U85 

324J2 

"3707708 

34433 

2.9J421 

Ik) 

1 

88706 

3.48359 

30605 

3.2(174.-, 

32524 

3.07I04 

34465 

2.90147 

59 

o 

28738 

3.47977 

3(11137 

3.2.1M,, 

32556 

3.071(10 

34498 

2.89873 

58 

3 

28763 

3.4759 

305(59 

3.28061 

32583 

3.06851 

3  1530 

2.89600 

57 

4 

28800 

3.4721(5 

30700 

3.25729 

32(521 

3.0(5(554 

34583 

2.893-27 

56 

5 

2883-2 

3.4W3< 

30732 

3.25392 

32653 

3.0;;-2.-)2 

34596 

2.89055 

55 

(j 

288  54 

3.4(5458 

307(54 

3.25055 

32685 

3.0595,. 

34(528 

2.88783 

54 

23895 

3.4GiHn 

30796 

3.5&4719 

32717 

3.05(149 

31(1(51 

•2.  88.")  11 

53 

8 

28927 

3.457U3 

30828 

3.24383 

32749 

3.05349 

341193 

2.  ,88240 

52 

9 

28938 

3.45327 

30860 

3.24049 

32782 

3.05049 

34786 

2.87970 

51 

10 

28930 

3.44951 

30891 

3.  237  14 

32814 

3.04749 

34758 

2.877011 

50 

11 

26031 

3.4457(5 

30923 

3.23381 

34846 

3.0445!! 

34791 

2.87430 

49 

12 

29053 

3.44202 

30955 

3.23048 

32878 

3.04152 

31824 

2.87161 

48 

13 

29084 

3.43S29 

30987 

3.u27  15 

329J1 

3.03-51 

34856 

2.86892 

47 

14 

•29111) 

3.4345(5 

3  I  ill  9 

3.22384 

32943 

3.0355(1 

34889 

2.86624 

46 

15 

•29147 

3.430-M 

31051 

3.22053 

32975 

3.03200 

34922 

2.8li35;i 

45 

18 

29179 

3.4-2713 

31083 

3.21722 

33007 

3.02963 

34954 

2.86089 

44 

17 

29210 

3.42313 

31115 

3.2U92 

33040 

3.02661 

34987 

2.85822 

43 

18 

29042 

3.41973 

31  147 

3.21063 

33072 

3.02372 

35019 

2.85555 

42 

19 

29274 

3.41694 

31178 

3.20734 

33104 

3.02977 

35052 

2.85289 

41 

20 

29305 

3.41250 

31210 

3.2040!) 

3313;; 

3.01783 

35085 

2.  85923 

40 

21 

29337 

3.40819 

31242 

3.23079 

33169 

3.01489 

35117 

2.84758 

39 

•>-> 

293(5* 

3.4050-J 

31274 

3.I9752 

33201 

3.  (11  191- 

35150 

2.84494 

38 

2:1 

29400 

3.4013(5 

31306 

3.19426 

33233 

3.00903 

35183 

2.84229 

37 

24 

29432 

3.39771 

31338 

3.19100 

332  iti 

3.00(511 

35216 

2.83965 

36 

25 

29463 

3.3910.1 

31370 

3.18775 

33-298 

3.00319 

352  18 

2.83702 

35 

26 

29495 

3.39342 

31402 

3.18451 

33330 

3,00028 

35281 

2.83439 

34 

27 

295-2(5 

3.38879 

31434 

3.18127 

33353 

2.9973S 

35314 

2.83176 

33 

28 

23558 

3.  3  S3  17 

31166 

3.17804 

33395 

2.99447 

3534(5 

2.82914 

32 

29 

23593 

:!.37!i:>.-) 

31498 

3.17481 

33427 

2.9915H 

35379 

2.82453 

31 

30 

29,521 

3.37594 

31533 

3.17159 

33460 

2.98868 

35412 

2.82391 

30 

31 

•in.™ 

3.37234 

31562 

3.ir>838 

33492 

2.98580 

35445 

2.  82  139 

23 

3-2 

•21  X, 

3.315875 

31594 

3.1()-)I7 

33521 

2.9-29J 

35477 

2.81870 

28 

33 

29716 

3.3(5510 

31(52(5 

3.1(5197 

33557 

2.98004 

35510 

2.81610 

27 

34 

23748 

3.311.-,- 

31(558 

3.13877 

335-i  9 

2.97717 

35543 

2.81350 

26 

35 

29780 

3.3.V30 

31690 

3.]  5.-,58 

3:M21 

2.9743d 

35576 

2.81091 

25 

30 

23811 

3.3.1443 

32722 

3.15210 

33654 

2.97144 

35608 

2.86633 

24 

37 

•29-  H 

3.:r,iH7 

31754 

3.14.122 

33  i-  i 

2.  96858 

35641 

2.80574 

23 

33 

•29-75 

3.3473-2 

3178(5 

:i.  ii  ;:).', 

33718 

2.96573 

351574 

2.80316 

22 

39 

29906 

3.34371 

31818 

3.14288 

33751 

2.9:5288 

35797 

2.80059 

21 

40 

29338 

3.31:1-2:! 

bl850 

3.1397-2 

33783 

2.  OlOOl 

35740 

•2.79-*02 

20 

41 

29970 

3.  3  Hi  70 

31882 

3.13356 

33816 

2.9572) 

35772 

2.79545 

19 

4-2 

300")  1 

3.33317 

31914 

3.  133  II 

33848 

2.95437 

35805 

2.79289 

18 

4:{ 

:',o  m 

3.:i-2i)i:, 

31946 

3.  13)27 

33831 

-2.9515.-, 

2.79  133 

17 

44 

311  M15 

3.32(514 

31978 

3.1-2713 

33913 

2.94872 

35871 

2.78778 

16 

45 

30097 

3.3-22I54 

32010 

3.12400 

33945 

2.94593 

35934 

2.78523 

15 

4(5 

39128 

3.31914 

32042 

3.12037 

33978 

2.94309 

35937 

2.78269 

14 

47 

3')  l!i() 

3.31585 

32074 

3.11775 

3!  110 

2.94)28 

35999 

2.78014 

13 

48 

3019-2 

3.31-21(5 

3-210(5 

3.114'14 

34  143 

2.93748 

3(1092 

2.777(51 

12 

49 

30224 

3.30868 

32139 

3.11153 

34075 

2.93468 

36035 

2.77507 

11 

50 

30255 

3.30521 

32171 

3.10812 

34108 

2.93UM 

3150(58 

-2.7:2:,! 

10 

51 

30287 

3.30174 

322.13 

3.10532 

34140 

2.  929  In 

3(1101 

2.77002 

9 

5-2 

30319 

3.29829 

32235 

3.19223 

34173 

2.92:132 

3(5134 

2.7(1759 

8 

53 

30351 

3.29483 

322.17 

3.09914 

34205 

•2.92351 

36167 

2.7IM9S 

7 

54 

3  i3~'2 

3.29139 

3i299 

'{  09:50(5 

34238 

2.92076 

33199 

2.76247 

6 

55 

30414 

3.28795 

32331 

3.092.1-' 

34270 

2.91799 

3623-2 

2.75996 

5 

56 

3.1441 

3.-2H1.Y.' 

323i53 

3.08:191 

3  1303 

•2.  '91  523 

382H5 

2.7574(5 

4 

57 

30478 

3.38109 

33396 

34335 

2.91-2II1 

30298 

2.  75  1  9(5 

3 

58 

3350.) 

3,27767 

3.08379, 

34368 

•2.9.1971 

3033J 

2.7521:5 

2 

59 

30341 

3.274-2!) 

32450 

3.08073 

3  MOO 

2.'»)f.9' 

3(13  '4 

•2.74997 

1 

(ii) 

80573 

3.27.K-, 

32492 

3.07768 

34133 

2.90421 

36397      2.74748 

0 

if 

N.  Cot. 

N.  T;i» 

N.  T«.'i.  i 

X.  Cot. 

N.Tan. 

N.  Cot]  N.  Tan. 

~M~ 

73  Degrees. 

7'3  Degrees,    i 

71  Degrees. 

70  Degrees. 

78 


NATURAL    TANGENfS. 


20  Degrees. 

21  Degrees. 

22  Degrees.    23  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.Tan 

N.  Cot. 

N.Tan 

N.Cot. 

N.Tan.]  N.Cot.  j  M 

~0 

36397 

2.74748 

38386 

2.60509 

40403 

2.475U9 

42447 

2.35585  60 

1 

3(5430 

2.744U9 

38420 

2.60283 

40436 

2.47302 

4248-2 

2.35395  59 

2 

36463 

2.74251 

38453 

2.60057 

40470 

2.47095 

4-2516 

2.35-205 

58 

3 

36496 

2.74004 

38487 

2.59831 

40504 

2.46888 

42551 

2.35015 

57 

4 

36529 

2.73756 

385-20 

2.59600 

40538 

2.46682 

42585 

2.34825 

5(5 

5 

36562 

2.73509 

38553 

2.59381 

40572 

2.46476 

42619 

2.3463b 

55 

6 

36595 

2.73263 

38587 

2.59156 

40606 

2.46270 

42054 

2.34447 

54 

7 

361.28 

2.73017 

38620 

2.58932 

40640 

2.46085 

42688 

2.34258 

53 

8 

36fi61 

2.72771 

38654 

2.58708 

40074 

2.45860 

42722 

2.34069 

52 

9 

36694 

2.72526 

38687 

2.58484 

40707 

2.45655 

42757 

2.33881 

51 

10 

36727 

2.72-281 

38721 

2.58261 

40741 

2.45451 

42791 

2.33693 

50 

11 

36760 

2.72036 

38754 

2.58038 

40775 

2.45246 

42826 

2.33505 

49 

12 

36793 

2.71792 

38787 

2.  57815 

40809 

2.45043 

42860 

2.33317 

48 

13 

36826 

2.71548 

38821 

2.57593 

40843 

2.44839 

42894 

2.33130 

47 

14 

36859 

2.71305 

38854 

2.57371 

40877 

2.44636 

42929 

2.32943 

46 

15 

36892 

2.71062 

38888 

2.57150 

40911 

2.44433 

42963 

2.32756 

45 

16 

36925 

2.70819 

38921 

2.56928 

40945 

2.44230 

42998 

2.32570 

44 

17 

36958 

2.70577 

38955 

2.56707 

40979 

2.440-27 

43032 

2.32383 

43 

18 

3699J 

2.70335 

38988 

2.56487 

41013 

2.43825 

43067 

2.32197 

42 

19 

37024 

2.70094 

39022 

2.56266 

41047 

2.43623 

43101 

2.32012 

41 

20 

37057 

2.69853 

39055 

2.56046 

41081 

2.43422 

43136 

2.31826 

40 

21 

37090 

2.69612 

39089 

2.55827 

41115 

2.43220 

43170 

2.31641 

39 

22 

37123 

2.69371 

39122 

2.55608 

41149 

2.43019 

43205 

2.31456 

38 

23 

37157 

2.69131 

39156 

2.55389 

41183 

2.42819 

43239 

2.31271 

37 

24 

37190 

2.68892 

39190 

2.55170 

41217 

2.42618 

43274 

2.31080 

36 

25 

37223 

2.68653 

39223 

2.54952 

41251 

2.42418 

43308 

2.30902 

35 

26 

37256 

2.68414 

39257 

2.54734 

41285 

2.42218 

43343 

2.30718 

34 

27 

37289 

2.68175 

39290 

2.54516 

41319 

2.42019 

43378 

2.30534 

33 

28 

37322 

2.67937 

39324 

2.54299 

41353 

2.41819 

43412 

2.30H5J 

32 

29 

37355 

2.67700 

39357 

2.54082 

41387 

2.41620 

43447 

2.30167 

31 

30  . 

37388 

2.67462 

39391 

2.53865 

41421 

2.41421 

43481 

2.29984 

30 

31 

37422 

2.67225 

39425 

2.53t>48 

41455 

2.41223 

43516 

2.29801 

29 

32 

37455 

2.66989 

39458 

2.53432 

41490 

2.41025 

43550 

2.29619 

28 

33 

37488 

2.66752 

39492 

2.53217 

41524 

2.40827 

43585 

2.29437 

27 

34 

37521 

2.66516 

39526 

2.53001 

41558 

2.40629 

43620 

2.29254 

26 

35 

37554 

2.66281 

39559 

2.52786 

41592 

2.40432 

43(554 

2.29073 

25 

36 

37588 

2.66046 

39593 

2.52571 

41626 

2.40235 

43689 

2.28891 

24 

37 

37621 

2.65811 

39626 

2.52357 

41660 

2.40038 

43724 

2.28710 

23 

38 

37654 

2.C5576 

39660 

2.52142 

41694 

2.39841 

43758 

2.285-28 

22 

39 

37687 

2.65342 

39694 

2.51929 

41728 

2.39045 

43793 

2.28348 

21 

40 

37720 

2.65109 

39727 

2.51715 

41703 

2.39449 

43828 

2.28167 

20 

41 

37754 

2.64875 

39761 

2.  51502 

41797 

2.39253 

438(52 

2.27987 

19 

42 

37787 

2.64642 

39795 

2.51289 

41831 

2.39058 

43897 

2-27806 

18 

43 

37820 

2.61410 

39829 

2.51076 

41865 

2.38862 

43932 

2.27026 

17 

44 

37853 

2.64177 

39862 

2.50864 

41899 

2.38668 

43966 

2.27447 

16 

45 

37887 

2.63945 

30896 

2.50652 

41933 

2.38473 

44001 

2.27267 

15 

4f> 

37920 

2.63714 

39930 

2.50440 

41968 

2.38279 

44036 

2.27088 

14 

47 

37953 

2.63483 

39963 

2.50229 

42002 

2.38084 

44071 

2.20909 

13 

48 

37986 

2.63252 

39997 

2.50018 

42036 

2.37891 

44105 

2.26730 

12 

49 

38020 

2.63021 

40031 

2.49807 

42070 

2.37697 

44140 

2.26552 

11 

50 

38053 

2.62791 

40065 

2.49597 

42105 

2.37504 

44175 

2.26374 

10 

51 

38086 

2.62561 

40098 

2.49386 

42139 

2.37311 

44210 

2.26196 

9 

52 

38120 

2.62332 

40132 

2.49177 

42173 

2.37118 

44244 

2.20018 

8 

53 

38153 

2.62103 

40166 

2.48967 

42207 

2.36925 

44279 

2.25840 

7 

54 

38186 

2.61874 

40200 

2.48758 

4-2242 

2.36733 

44314 

2.25663 

6 

55 

38220 

2.61646 

40234 

2.48549 

42276 

2.36541 

44349 

2.25486 

5 

56 

38253 

2.61418 

40267 

2.48340 

42310 

2.36349 

44384 

2.25309 

4 

57 

38-286 

2.61190 

40301 

2.4813-2 

42345 

2.36158 

44418 

2.25132 

3 

58 

38320 

2.  609(53 

40335 

2.47924 

42379 

2.359(57 

44453 

2.24956 

2 

59 

38353 

2.60736 

40369 

2.47716 

42413 

2.35776 

44488 

2.24780 

1 

60 

38386 

2.60509 

10403 

2.47509! 

42447 

2.35585 

44523 

2.24604 

0 

M 

N  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan. 

M 

69  Degrees. 

68  Degrees. 

67  Degrees. 

66  Degrees. 

NATURAL  TANGENTS. 


79 


24  Degrees. 

25  Degrees. 

26  Degrees. 

27  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N  T;.n. 

L\.     Cllt. 

N.  Tan 

N.  Cot. 

M 

0 

44523 

2.24lU)4 

48631 

2.14451 

48773 

2.05030 

50953 

.'.H5261 

(50 

1 

44596 

2.244'28 

46666 

•2.  I428H 

48809 

2.04879 

50989 

.'.Mi  1-20 

59 

44593 

2.24252 

46702 

2.141-25 

48845 

2.04728 

51036 

.95979 

58 

3 

44687 

•2.  -2H>77 

4(i737 

2.139ii3 

48881 

2.04577 

51063 

.95838 

57 

4 

44662 

2.  23902 

46772 

2.  13801 

48917 

•2.041-2ii 

51H99 

.951)98 

56 

5 

44(597 

2  .  -23727 

46608 

2.13(539 

48953 

-2.04276 

51136 

.95557 

55 

6 

44739 

2.23553 

4(5843 

2.13477 

48989 

2.04125 

51173 

.95417 

54 

7 

44767 

•2.2337H 

46879 

a.  133  1  6 

49096 

2.03975 

51909 

.95277 

53 

8 

4480-2 

2.  -23204 

46914 

2.13154 

49062 

2.0:1825 

51246 

.95137 

52 

9 

44837 

a.  23030 

46950 

•2.  |-2993 

49068 

2.03675 

51283 

.94!I97 

51 

10 

44872 

2.-2-2rtr>7 

46985 

2.12832 

49134 

2.03526 

51319 

.94858 

50 

11 

44907 

•2.  -2-26*3 

47021 

2.12671 

49170 

2.03376 

51356 

.94718 

49 

19 

44942 

2.22510 

47056 

2.12511 

49906 

2.03227 

51393 

-94579 

48 

13 

44!I77 

-2.-22.t37 

47092 

2.J2350 

49242 

2.03078 

51430 

.94440 

47 

14 

45012 

2.22164 

47128 

2.12190 

49278 

2.02929 

51467 

.94301 

46 

15 

45047 

2.21992 

47163 

2.12030 

49315 

2.02780 

51503 

.94162 

45 

Ifi 

45082 

2.21819 

47199 

2.11871 

49351 

2.02631 

51540 

.94023 

44 

17 

45117 

2.21647 

47234 

2.11711 

49387 

2.02483 

51577 

.93885 

43 

18 

45152 

2.21475 

47270 

2.11552 

49423 

2.02335 

51614 

.93746 

42 

19 

45187 

2.21304 

47305 

2.11392 

49459 

2.02187 

51651 

.93608 

41 

20 

45222 

2.21132 

47341 

2.11233 

49495 

2.02039 

51688 

.93470 

40 

21 

45257 

2.209(51 

47377 

2.11075 

49532 

2.01891 

51724 

.93332 

39 

22 

45-292 

2.20790 

47412 

2.10916 

495(58 

2.01743 

51761 

.93197 

38 

33 

453-27 

2.20619 

47448 

2.10758 

496(14 

2.01596 

51798 

.93057 

37 

24 

45362 

2.20449 

47483 

2.10(500 

49640 

-2.01449 

51835 

.  92920 

36 

25 

45397 

•2.  -20-278 

47519 

-2.10441 

49677 

2.01302 

51872 

.92782 

35 

26 

45432 

2.20108 

47555 

2.10284 

49713 

2.01155 

51909 

.92645 

34 

27 

45467 

2.  Hi:  138 

47590 

2.10126 

49749 

2.01008 

51946 

.92508 

33 

28 

45502 

2.19769 

47(526 

2.09969 

49786 

2.00862 

51983 

.92371 

32 

99 

45537 

2.  19599 

47(562 

2.09811 

49822 

2.00715 

52020 

.92235 

31 

30 

45573 

2.19430 

47698 

2.09654 

49856 

2.00569 

52057 

.92098 

30 

31 

45608 

2.19261 

47733 

2.09498 

49894 

2.00423 

52094 

.91962 

29 

3-2 

45643 

2.191192 

47769 

2.09341 

49931 

2.00277 

52131 

.91*25 

28 

33 

45678 

2.18923 

47895 

2.09184 

49967 

2.00131 

-  52168 

.91690 

27 

34 

45713 

2.18755 

47840 

2.09028 

50004 

1.99986 

52205 

.91554 

26 

35 

45748 

2.18587 

47876 

2.08-!72 

50040 

l-'.i'.H41 

59242 

.91418 

25 

3!5 

45784 

2.18419 

47912 

•2.0S7I6 

58076 

1.9!  16:  (5 

52378 

.91282 

!24 

37 

45819 

•2.  lr-251 

47948 

2.08560 

50113 

I.  <  >9550 

59316 

•91148 

23 

38 

45854 

2.18084 

47984 

2.08405 

50149 

1.  '.19406 

5-2353 

.<)10I7 

22 

39 

45889 

2.17916 

48019 

2.08250 

50185 

1.99261 

52390 

-90876 

21 

40 

45924 

2.  17749 

•181)55 

•2.<H!)'.M 

50222 

1.99116 

52427 

.90741 

20 

41 

459!50 

3.  17582 

48091 

2.07939 

50258 

1.98972 

52464 

.90607 

19 

42 

45905 

2.17410 

48127 

2.07785 

50295 

1-98828 

52501 

.90472 

18 

43 

46030 

2.17249 

48163 

2.07630 

50331 

1.98684 

52538 

.9)337 

17 

44 

46065 

2.17083 

48198 

2.07476 

503(58 

1.98540 

52575 

.90203 

16 

45 

46101 

2.16917 

48234 

2.07321 

50404 

1.98396 

52613 

.90069 

15 

4(5 

46136 

2.16751 

48270 

2.07167 

50441 

1.98253 

52650 

.89935 

14 

47 

4C.171 

2.  16585 

4830(5 

2.07014 

50477 

1.98110 

5-26S7 

.89801 

13 

48 

46206 

2.16420 

48342 

2.  0^)860 

50514 

1.97'.)6i> 

52724 

.89667 

12 

49 

46242 

2.1(1255 

48378 

2.0(5706 

50550 

1.978-2:1 

52761 

.89533 

11 

50 

40877 

2.16090 

48414 

2.06553 

50587 

1.97680 

5*2798 

.89400 

10 

51 

463U 

2.  15925 

48450 

2.0')400 

5015-23 

1.97538 

52836 

.89-266 

9 

52 

415348 

2.  15760 

48486 

2.06247 

50660 

1.97395 

52873 

.89133 

8 

53 

46383 

•2.  I559li 

4H.V21 

2.0H094 

50596 

1.  97253 

52910 

.89'IO!> 

7 

54 

46418 

•2.15J32 

48557 

•2.0591-2 

50733 

1.9711] 

52947 

.88867 

6 

55 

46454 

2.152(58 

48593 

2.05789 

507(59 

1-969(59 

52984 

.88734 

5 

5(5 

46489 

2.15104 

48629 

2.05637 

508*1 

1.96827 

53024 

.88602 

4 

57 

46525 

2.14940 

48(565 

2.05485 

50843 

1.96685 

53059 

.88469 

3 

58 

46540 

2.14777 

48701 

L>.05.->33 

50879 

1.96544 

53096 

.88337 

2 

59 

41  J595 

2.14614 

48737 

2.0518-2 

509115 

1.9(i4D-2 

53134 

.88205 

1 

60 

46631 

2.14451 

48773 

2.05030 

50953 

1.96261 

53171 

.88073 

0 

M 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot.  IN.  Tan. 

N.  Cot.   N.  Tan. 

M 

65  Degrees. 

64  Degrees. 

63  Degrees. 

62  Degrees. 

80 


NATURAL  TANGENTS. 


28  Degrees. 

29  Degrees. 

30  Degrees.    31  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.Tan 

IS'.  Cot. 

N.Tai.  j  N.Cot.  N.Tan 

N.  Cot.   M  i 

0 

53171 

.88073 

55431 

1.80405 

57735 

.73205  60086 

1.66428  60 

1 

53208 

.87941 

55469 

.80281 

57774 

.73089  60120 

1.  00318  59 

2 

53240 

.87809 

55507 

.80158 

57813 

.72973  G0165 

1.06-201) 

58 

3 

53-283 

.87677 

55545 

.80034 

57851 

.7-2f'57  OU-2U5 

1.60099 

57 

4 

53320 

.87546 

55583 

.79U11 

57890 

.72741   r,«f245 

1.65990 

56 

5 

53358 

.87415 

55621 

.79788 

57929 

.7-2ii-,T)  :  60284 

1.65881 

55 

6 

53395 

.87283 

55659 

.79665 

57-J68 

.7-2509  1  103-24 

1.65772 

54 

7 

53432 

.87152 

55697 

.79542 

58007 

.72393  60304 

1.65663 

53 

8 

53470 

.87021 

55736 

.79419 

58046 

.72278  00403 

1.65554 

52 

9 

53507 

.86891 

55774 

.79296 

58085 

.7-2103  00443 

1.65443 

51 

10 

53545 

.86760 

55812 

.79174 

58124 

.72047  i  60483 

1.65337 

50 

11 

5358-2 

.86630 

55850 

.79051 

58162 

.71932  60522 

1.65986 

49 

12 

53620 

.86499 

55888 

.78929 

58-201 

.71817  60562 

1-05120 

48 

13 

53657 

.86369 

5592(5 

.78807 

58240 

.71702  6(1002 

1.05011 

47 

14 

53694 

.86239 

56964 

.78685 

58279 

.71588 

60642 

1.04903 

46 

15 

53732 

.86109 

56003 

.78563 

58318 

.71473 

60681 

1.64785 

45 

16 

53769 

.85979 

56041 

.78441 

58357 

.71358 

60721 

1.64687 

44 

17 

53807 

.85850 

56079 

.78319 

58396 

.71244 

60761 

1.64579 

43 

18 

53844 

.85720 

56117 

.78198 

58435 

.71129  OOSlll 

1.64471 

42 

19 

53882 

.85591 

56156 

.78077 

58474 

.71015 

60841 

1.04363 

41 

20 

53920 

.85402 

56194 

.77955 

58513 

.70901 

60881 

1.64256 

40 

21 

53957 

.85333 

56232 

.77834 

58552 

.70787 

60921 

1.64148 

39 

22 

53995 

.85204 

56270 

.77713 

58591 

.70673 

60960 

1.64041 

38 

23 

54032 

.85075 

56309 

.77592 

58631 

.70560  j  61000 

1.03933 

37 

24 

54070 

.84946 

56347 

.77471 

58670 

.70446  |  61040 

1.6382(5 

36 

25 

54107 

.84818 

56385 

.77351 

58709 

.70332 

61080 

1.63719 

35 

•26 

54145 

.84689 

564-24 

.77230 

58748 

.70219 

61120 

1.63612 

34 

27 

54183 

•84561 

56462 

.77110 

58787 

.70106 

61100 

1.63505 

33 

28 

54220 

.84433 

56500 

.76990 

58826 

.69992 

61200 

1.03398 

32 

29 

54258 

.84305 

56539 

.76869 

58865 

.69879 

61240 

1.63292 

31 

30 

54296 

.84177 

56577 

.7674!» 

58904 

.69766 

01  .'80   1.63185 

30 

31 

54333 

.84049 

56616 

.76629 

58944 

.69653 

61320 

1.63079 

29 

3-2 

54371 

.839-22 

56654 

.76510 

58983 

.69541 

61360 

1.03972 

28 

33 

54460 

.83794 

56693 

.703SIO 

59022 

.69428 

61400 

1.62866 

27 

34 

5444!! 

.83667 

56731 

.76271 

59061 

.6i»315 

61440 

1.62700 

20 

!  3.5 

54484 

.83540 

56769 

-7615] 

59101 

.69203 

61480 

1  62054 

25 

36 

54522 

.83413 

56808 

.76032 

59140 

.69091 

61520 

1.62548 

24 

37 

545fiO 

.83286 

56846 

.75913 

59179 

.68979 

61561 

1.62442 

23 

38 

54597 

.83159 

56885 

-75794 

59218 

.68866 

61601 

1.62336 

22 

39 

54635 

.83033 

56923 

.75675 

59258 

.68754 

61641 

1.62230 

21 

40 

54673 

.82906 

57982 

.75556 

59297 

.68643 

61681 

1.62125 

20 

41 

54711 

.82780 

57000 

.75437 

59336 

.68531 

61721 

1.62019 

19 

42 

51748 

.82654 

57039 

.75319 

59376 

.68419 

S1761 

1.61914 

18 

43 

54786 

.82528 

57078 

.75200 

59415 

.68308 

61801 

1.61808 

17 

44 

54824 

.82402 

57116 

.75082 

59454 

.68196 

61842 

1.61703 

16 

45 

54862 

.82276 

57155 

.74964 

59494 

.68085 

61882 

1.61598 

15 

46 

54900 

.82150 

57193 

.74846 

59533 

.07974 

61922 

1.61493 

14 

47 

54933 

.82025 

57232 

.74728 

59573 

.67968 

61962 

1.61388 

13 

48 

54975 

.81899 

57271 

.74610 

59612 

.67752 

6-2003 

1.61283 

12 

49 

55013 

.81774 

57309 

.74492 

59651 

.67641 

62043 

1.61179 

11 

50 

55051 

.81649 

57348 

.74375 

59691 

.67530 

62083 

1.61074 

10 

51 

55088 

.81524 

57386 

.74257; 

59730 

.67419 

62124 

1.60970 

9 

5'2 

55127 

.81399 

57425 

.74140J 

59770 

.67309 

62164 

1.60865 

8 

53 

55165 

.81274 

57464 

.74022 

59809 

.67198 

62204 

1.60761 

7 

54 

55903 

.81150 

57503 

.739(15 

59849 

67088 

69345 

1.60657 

6 

55 

55241 

.81025 

57541 

.73788 

59888 

66978; 

6-2-285 

1.60553 

5 

56 

55279 

.80901 

57580 

.73671 

59928 

668671 

6-2325 

1.60449 

4 

57 

55317 

.80777 

57619 

.73555 

59967 

66757 

62366 

1.60345 

3 

58 

55355 

.80653 

57657 

.73-138 

60007 

66647 

62406 

1.60241 

2 

59 

55393 

.80529 

57696 

.73321 

60046 

.66538 

62446 

1.60137 

1 

60 

55431 

.80405 

57735 

1.73205 

60086 

.66428 

62487 

1.60033   0 

M 

N  Cot. 

N.  Tan 

N.  Cot. 

N.  Tan. 

N.  Cot.  N.  Tan. 

N.  Cot. 

N.  Tan. 

M 

61  Degrees. 

GO  Degrees.  i|  59  Degrees. 

58  Degrees. 

NATURAL  TANGKNT-S. 


32  Detrn  •••<. 

33  Degrees. 

34  Degrees. 

35  Degrees. 

M 

N.Tan. 

N.  Cot. 

M.Taa 

A.  Cot. 

.VTan 

N.  Cot. 

N.Ta  ii 

N.  Cot. 

M 

0 

02487 

1.  60D33 

64941 

i.  53981 

674.51 

1.48351 

70021 

1.42815 

60 

1 

6-2527 

1.5'.I93( 

64983 

1.538-^ 

67493 

1.4816: 

70064 

1.437ft) 

59 

2 

62568 

I..V.H-2! 

65033 

L.5379 

67538 

1.48074 

70107 

1.42638 

58 

:i 

63808 

I.5975E 

65065 

1.53693 

67578 

1.47977 

70151 

1  .48554 

57 

4 

6-2649 

L.  59691 

65106 

1  .  535S»: 

67620 

[.4788J 

70194 

1.  434ft 

56 

5 

82689 

1.59517 

<;;>)  48 

1.53497 

67883 

1.47793 

70238 

1.4-2374 

55 

i    l! 

62730 

1.  r,;)l  14 

65189 

l.5340( 

6770.) 

1  ,47699 

70281 

1.  43381 

54 

7 

(i-2770 

i.f»93ii 

65231 

1.5330, 

87748 

1.47607 

70325 

1.42198 

53 

8 

62811 

1.59908 

65273 

1.53205 

67790 

1.47514 

70368 

1.42111 

52 

9 

62852 

L.59I05 

65314 

1.53107 

67832 

1.474  '22 

70112 

1.43038 

51 

10 

62892 

1.59002 

6.53.55 

1.530K 

67875 

1.4733(1 

70455 

1.41934 

50 

11 

62933 

1.58901 

65397 

I,fr2'.li: 

671117 

1.47338 

70  MW 

1.41847 

49 

12 

62973 

1.587W 

65438 

1.538K 

67960 

1.47146 

7054-2 

1-41750 

48 

13 

63014 

I.596KS 

65480 

1.53711 

68009 

1.47053 

70586 

1.41673 

47 

11 

153.  )f>3 

1.5859: 

65531 

I.52C.2:, 

68045 

1.46982 

70629 

1.41584 

4G« 

15 

63095 

i.r>Hioa 

65563 

I..  5-2.5-2: 

68088 

1.46870 

701573 

1  .41497 

45 

i    1:5 

63134 

1.58388 

(55604 

1..  5-242! 

68130 

1.46778 

70717 

1.41409 

44 

17 

63177 

1.58386 

65646 

L.  52339 

68173 

1.46686 

70760 

1.41333 

43 

18 

63-217 

1.58184 

65688 

1.52235 

68215 

1.46595 

70804 

1.41235 

42 

19 

63258 

1.581)83 

6572!) 

I.52I3<» 

68258 

1  .46503 

70848 

1.41148 

41 

•20 

0329;) 

1.57981 

65771 

1.52043 

68301 

1.  4(1411 

70891 

1.41061 

40 

21 

63340 

1.5787!) 

65813 

J.  51946 

68343 

1.46329 

7093.5 

1.40974 

39 

2-2 

63380 

1.57778 

65854 

1.51850 

(58386 

1.46339 

70979 

1.40867 

38 

23 

63421 

1.57676 

65HUO 

1.51754 

68429 

1.46137 

71023 

1.40800 

37 

•2t 

63463 

1.57575 

(55938 

1.51658 

68471 

1.4(504(5 

71066 

1.40714 

36 

25 

63503 

1.57474 

65980 

1.51562 

68514 

1.45955 

71110 

1.40(527 

35 

26 

G3544 

1.57372 

86021 

1.51466 

68557 

1.458(54 

71154 

1.40540 

34 

27 

63594 

1.57271 

66063 

1.51370 

68600 

1.45773 

71198 

1.40454 

33 

28 

63635 

1-57170 

66105 

1.51275 

681542 

1.45682 

71242 

1.40368 

32 

29 

63996 

1.57069 

66147 

1.51179 

68685 

1.45593 

71285 

1.40281 

31 

30 

63707 

1.56969 

66189 

1.51084 

68788 

1.45501 

71329 

1.40195 

30 

31 

81748 

1.56863 

66230 

1.50988 

6S771 

1.45410 

71373 

1.40109 

29 

3-2 

63789 

1.56767 

66272 

1.50893 

68814 

1.453-20 

71417 

1.40022 

28 

33 

63830 

1.5(5667 

66314 

1-50797 

68857 

1.4522!) 

71461 

1.39936 

27 

34 

(53871 

1.56566 

66356 

1.50702 

68900 

1.45139 

71505 

1.39850 

26 

35 

6391-2 

I.5ti46)i 

66398 

1.50607 

(58942 

1.45048 

71549 

1.39764 

25 

36 

63953 

1.56366 

66440 

1.50512 

88  185 

1.449.58 

71593 

1.39679 

24 

37 

63994 

l.5->-2f>5 

C.'i  IH-2 

1.50417 

69028 

1.44868 

71637 

1.39.593 

23 

38 

64035 

1.56165 

66524 

1.5032-2 

69071 

1.44778 

71681 

1.39507 

22 

39 

6407!> 

1.56085 

66566 

L.  50328 

69114 

1  .44688 

71725 

1.39421 

21 

40 

64117 

l.:>:>%6 

66008 

1.50133 

69157 

1  .  1  1.51H 

7171!!) 

1.39336 

20 

41 

61153 

i.r>.->8!)6 

66550 

1.50038 

69200 

1  .  i  r>iH 

71813 

1.39350 

19 

4-2 

64199 

i..-»37(i6 

f56<>92 

1.49!)44 

69243 

1.44418 

71857 

1.39165 

18 

43 

64240 

1.55666 

66734 

1.49819 

69-286 

1.44339 

71901 

L  39679 

17 

44 

64381 

1.55567 

86778 

1.41)755 

69329 

1  .  1  423SI 

71946 

1.38994 

16 

45 

64322 

1.55467 

66818 

1.49661 

69372 

1.44149 

71990 

1.38900 

15 

415 

64363 

1.55368 

66860 

1.49566 

69416 

1.44060 

7-2034 

i.:;--.-l 

14 

47 

64404 

1.55269 

66902 

1.49472 

69459 

1.43970 

72078 

1.38738 

13 

48 

64446 

1.55170 

66944 

1.49378 

(59502 

1.43881 

72122 

1.38653 

12 

49 

64487 

1.55071 

66986 

1.49284 

69545 

1.43792 

72166 

1.38568 

11 

50 

64528 

1.54972 

67028 

1.4919(1 

69588 

1.43703  1|  72211 

1.38484 

10 

51 

64569 

1.54873 

67071 

1.49097 

69631 

1.43614     73255 

1.38399 

9 

52 

64610 

1.51774 

67113 

1.49003 

69675 

1.43525 

72299 

1.38314 

8 

53 

64652 

1.54675 

67155 

1.48909 

6!)718 

1.43436 

72344 

I.3833U 

7 

54 

64693 

1.54576 

67197 

1.48816 

69761 

1-43347 

72388 

1.38145 

6 

55 

64734 

1.54478 

67239 

1.48722 

69804 

1.  432.58 

72432 

1.3801)0 

5 

56 

64773 

1.51379 

67282 

1.48639 

69847 

1-43169 

72477 

1.371)76 

4 

57 

84817 

1.54381 

67324 

J  .  48536 

<i9->!)l 

1.43080 

73591 

1.37891 

3 

58 

1.51  183 

67366 

1.48442 

69934 

1.429U2 

72565 

1.37807 

2 

59 

64899 

1.5  1085  j 

67409 

1.4334!) 

69977 

1.42903 

72(510 

1.37722      1 

60 

61941 

1.  5:»9S(i 

67451 

1.48256 

70021 

1.42815 

72854 

1.37638      0 

M 

N    Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan 

N.  Cot 

N.  Tan. 

N.  Cot. 

N.Tan.    M 

57  Degrees. 

56  Degrees. 

55  Degrees. 

54  Degrees. 

4* 


8-2 


NATURAL  TANGENTS. 


M 

36  Degrees.  ||  37  Degrees. 

38  Degrees. 

39  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.  Tan. 

X.  Cot. 

i\?.  Tun. 

N.  Cot. 

N.  Tan 

N.  Cot. 

0 

72654 

.37638 

75355 

1.32704 

78129 

.27994 

8o978 

.23489 

60 

1 

72699 

.37554 

75401 

,3-2(124 

78175 

.27917 

81U27 

1.23411.; 

59 

2 

72743 

.37470 

75447 

.32544 

78832 

.27841 

81075 

.23343 

58 

3 

72788 

.37386 

75492 

.32464 

78269 

.27764 

81123 

.23270 

57 

4 

72832 

.37302 

75538 

.32384 

7*310 

.27688 

81171 

.23196 

56 

5 

72877 

.37218 

75584 

.32304 

78363 

.27011 

81220 

.23123 

55 

6 

72921 

.37134 

75629 

.32224 

78410 

.27.535 

81268 

.23050 

54 

72966 

.37050 

7f>07.-> 

.3-2144 

78457 

.27458 

81316 

.2-2977 

53 

8 

73010 

.36967 

75721 

.32064 

78504 

.27382 

81304 

.22904 

52 

9 

73055 

.308H3 

75767 

.31984 

78551 

.27306 

81413 

.22831 

51 

10 

73100 

.36800 

75812 

.31904 

78598 

.27230 

81461 

.2-2758 

50 

11 

73J44 

,36716 

75858 

.31825 

78645 

.27153 

81510 

.22085 

49 

12 

73189 

.36633 

75904 

.31745 

78692 

.27077 

81558 

.22612 

48 

13 

73234 

.36549 

75950 

.3l(>6(> 

78739 

.27001 

81606 

.22539 

47 

14 

73278 

.36406 

75996 

.31586 

78786 

.26925 

81655 

.22467 

46 

15 

73323 

.36383 

76042 

.31.507 

78834 

.26849 

81703 

.22394 

45 

16 

73368 

.36300 

76088 

.31427 

78881 

.26774 

81752 

.22321 

44 

17 

73413 

.36217 

78134 

.31348 

78928 

.26698 

81800 

.22249 

43 

18 

73457 

.36133 

76180 

.31269 

78975 

.26(522 

81849 

.22176 

42 

19 

73502 

.36051 

76226 

.31190 

79022 

.26540 

81898 

.22104 

41 

20 

73547 

.35968 

76272 

.31110 

79070 

.26471 

8194(5 

.22031 

40 

21 

73592 

.35885 

76318 

.31031 

79117 

.26395 

81995 

.21959 

39 

22 

73637 

.35802 

76364 

.30952 

79164 

.26319 

82044 

.21886 

38 

23 

73681 

.35719 

76410 

.30873 

79212 

.26244 

82092 

.21814 

37 

24 

73726 

.35637 

76456 

.30795 

79-259 

.26169 

82141 

.21742 

36 

25 

73771 

.35554 

76502 

.30716 

79306 

.26093 

82190 

.21670 

35 

26 

73816 

.35472 

76548 

.30637 

79354 

.26018 

82238 

.21598 

34 

27 

73816 

.35389 

76594 

.30558 

79401 

.25943 

82287 

.21526 

33 

28 

7390ti 

.35307 

76640 

.30480 

79449 

.25867 

82336 

.21454 

32 

29 

73951 

.352-24 

76686 

.30401 

79496 

.25792 

82385 

.21382 

31 

30 

73996 

.35142 

76733 

.30323 

79544 

.25717 

82434 

.21310 

30 

31 

74041 

.35060 

76779 

.30-244 

79591 

.25642 

8-2483 

.21238 

29 

32 

74086 

.34978 

76825 

.3016(i 

791)39 

.25567 

82531 

.21166 

28 

33 

74J31 

.34896 

76871 

.30087 

791)80 

.25492 

82580 

.21094 

27 

34 

74176 

.34814 

76918 

.30009 

79734 

.25417 

82629 

.21023 

26 

35 

74221 

.34732 

76964 

.29931 

79781 

.25343 

82678 

.20951 

25 

36 

74267 

.34650 

77010 

.29853 

79829 

.25268 

82727 

.20879 

24 

37 

74312 

.345t<8 

77057 

.29775 

79877 

.25193 

82776 

.20808 

23 

38 

74357 

.34487 

77103 

.39696 

799-24 

.25118 

82825 

.20736 

22 

39 

74402 

.34405 

77149 

.29618 

79972 

.25044 

82874 

.20665 

21 

40 

74447 

.34323 

771  96 

1.29541 

80020 

.24969 

82921} 

.20593 

20 

41 

74492 

.34242 

77242 

.29403 

80067 

-24895 

82972 

.20522 

19 

42 

74538 

.34160 

77289 

.29385 

80115 

.24820 

83022 

.20451 

18 

43 

74583 

.34079 

77335 

.29307 

80163 

.24740 

83071 

.20379 

17 

44 

74628 

.3399,- 

77382 

.29229 

80211 

.24672 

83120 

.20308 

16 

45 

74674 

.33916 

77428 

.2915'2 

80258 

.24597 

83169 

.20237 

15 

46 

74719 

.  33835 

77475 

.29074 

80306 

.24523 

83218 

.20166 

14 

47 

74764 

.33754 

77521 

.28997 

80354 

.24449 

83268 

.20095 

13 

48 

74810 

.33673 

77568 

.28919 

80402 

.24375 

83317 

.20024 

12 

49 

74855 

.33592 

77615 

.28842 

80450 

.24301 

83366 

.19953 

11 

50 

74900 

.  .33511 

77661 

.28764 

80498 

.24227 

83415 

-19882 

10 

51 

74946 

.33430 

77703 

.28687 

8054(5 

.24153 

83465 

.19811 

9 

52 

74991 

.33349 

77754 

.28610 

80594 

.24079 

83514 

.  19740 

8 

53 

75037 

.33268 

77801 

.28533 

80642 

.24005 

83564 

.19069 

7 

54 

75082 

.33187 

77H48 

.28456 

80690 

.23931 

83613 

.  19599 

6 

55 

75128 

.33107 

77895 

-28379 

80738 

.23858 

83602 

.  19528 

5  ' 

56 

75174 

.33021) 

77941 

.28302 

80786 

.23784 

83712 

.19457 

4 

57 

75219 

.32946 

77988 

.28225 

80834 

-23710 

83761 

.19387 

3 

58 

75264 

.32865 

77035 

.28148 

80882 

.23637 

83811 

.19316 

2 

59 

75310 

.32785 

77082 

1.28071 

80930 

.23563 

83860 

.19246 

1 

60 

75355 

1.32704 

77129 

1.27994 

80978 

.23490 

83910 

1.19175 

0 

"M" 

N.  Cot. 

N.Tan. 

N.  Cot.JN.  Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.  Tan. 

M 

53  Degrees. 

52  Degrees. 

51  Degrees. 

50  Degrees. 

NATtRAi.    TANCKNT*. 


40  Degrees. 

41  Degrees. 

4'2  Degrees. 

43  Degrees. 

M 

N.Tan. 

N.  Cot. 

N.  Tan. 

N.  Cot. 

N.Tan. 

N.  Cot. 

N.Tan. 

N.  Cot. 

M 

0 

83910 

1.19175 

86929 

1.15037 

901)40 

.11061 

93252 

.07237 

60 

1 

83960 

1.19105 

86980 

1.14909 

90093 

.  lUllllfi 

93306 

.07174 

59 

2 

84009 

1.19035 

87031 

1.14902 

90146 

.10931 

93360 

.07112 

58 

3 

84059 

1.18964 

87082 

1.14^34 

91)199 

.  108li7 

93115 

.07049 

57 

4 

84108 

1.18894 

87133 

1.14767 

90251 

.10802 

9154(19 

.06987 

56 

5 

84158 

1.18824 

87184 

1.14699 

90304 

.10737 

93524 

.()ti!>25 

55 

6 

84208 

1.18754 

87236 

1.14632 

90357 

.  10157-J 

93578 

.06862 

54 

7 

84258 

1.18684 

87287 

1.14565 

90410 

.10607 

93633 

.06800 

53 

8 

84307 

1.18614 

87338 

1.14498 

90463 

.  10543 

93688 

.06738 

52 

9 

84357 

1.18544 

87389 

1.J4430 

90516 

.10478 

93742 

.015670 

51 

10 

84407 

1.18474 

87441 

1.14363 

90569 

.10414 

93797 

.06613 

50 

H 

84457 

1.18404 

87492 

1.14296 

90(121 

.  10349 

93852 

.1111551 

49 

12 

84507 

1.18334 

87543 

1.1422!) 

90674 

.  102*-, 

93906 

.0(5-189 

48 

13 

8455G 

1.18264 

87595 

1.14162 

90727 

.10220 

93961 

.06427 

47 

14 

84606 

1.18194 

87646 

1.14095 

90781 

.10156 

94016 

.0(5365 

46 

15 

84656 

1.18125 

87698 

1.14028 

90834 

.10091 

94071 

.06303 

45 

16 

84706 

1.18055 

87749 

1.13961 

90887 

.10027 

94125 

.06241 

44 

17 

84756 

1.17986 

87801 

1.13894 

9094(1 

.091)63 

94180 

.06179 

43 

18 

84806 

1.17916 

87852 

1.13828 

90993 

.09899 

94235 

.06117 

42 

19 

84856 

1.17846 

87904 

1.13761 

91046 

.09834 

94290 

.oi;o:><; 

41 

20 

84906 

1.17777 

87955 

1.13694 

91099 

.09770 

94345 

.05994 

40 

21 

84956 

1.17708 

88007 

1.13627 

91  153 

.09706 

94400 

.05!  132 

39 

22 

85006 

1-17638 

88059 

1.13561 

91206 

.09642 

94455 

.05870 

38 

23 

85057 

1-17569 

88110 

1.13494 

91259 

.09578 

94510 

.05809 

37 

24 

85107 

1.17500 

88162 

1.13428 

91313 

.09514 

94565 

.05747 

36 

25 

85157 

1.17430 

88214 

1.13361 

91366 

.00450 

94620 

.05685 

35 

26 

85207 

1-17361 

88265 

1.13295 

91419 

.09386 

94(576 

.05(524 

34 

27 

85257 

1.17292 

88317 

1.13228 

91473 

.09322 

94731 

.0.-)5ti2 

33 

28 

85307 

1.17223 

88369 

1.13162 

91526 

.09258 

94786 

.05501 

32 

29 

85358 

1.17154 

88421 

1.13096 

91580 

.09195 

94841 

.05439 

31 

30 

85408 

1.17085 

88473 

1.13029 

91633 

.09131 

94896 

.05378 

30 

31 

85458 

1.17016 

88524 

1.12963 

91687 

.09067 

94952 

.05317 

29 

32 

85509 

1.16947 

88576 

1.12897 

91740 

.09003 

95007 

.05255 

28 

33 

85559 

1.16878 

88628 

1.12831 

91794 

.08940 

95062 

.05194 

27 

34 

85609 

1.16809 

88680 

1.12765 

91847 

.08876 

95118 

.05133 

26 

35 

85660 

1.16741 

88732 

1.12699 

91901 

.08813 

95173 

.05072 

25 

36 

85710 

1.16672 

88784 

1.12633 

91955 

.08749 

9522'J 

.05010 

24 

37 

85761 

1.16603 

88836 

1.12567 

92008 

.08686 

95284 

.0494!) 

23 

38 

85811 

1.16535 

88888 

1.12501 

92062 

.OdfrJ-J 

95340 

.04888 

22 

39 

85862 

1.16466 

88940 

1.12435 

92116 

.08559 

95395 

.04827 

21 

40 

85912 

1.16398 

88992 

1.  12369 

92170 

.0849(5 

95451 

.04766 

20  ! 

41 

85963 

1.16329 

89045 

1.12303 

92223 

.08432 

96506 

.0470;-) 

19 

42 

86014 

1.16261 

89097 

1.12238 

92277 

.08369 

95562 

.04644 

J8 

43 

86064 

1.16192 

89149 

1.12172 

92331 

.08306 

95618 

.04583 

17 

44 

86115 

1.16124 

89201 

1.12106 

92385 

.08243 

95673 

.04522 

1(5 

45 

86166 

1.16056 

89253 

1.12041 

92439 

.08179 

95729 

.04461 

15 

46 

86216 

1.15987 

89306 

1.11975 

92493 

.08116 

95785 

.04401 

14 

47 

86267 

1.15919 

89358 

1.11909 

92547 

.08053 

95841 

.04340 

13 

48 

86318 

1.15851 

89410 

1.11844 

92601 

.07990 

95897 

.0427H 

12 

49 

86368 

1.15783 

89463 

1.11778 

92655 

.07927 

96952 

.04218 

11 

50 

86419 

1.15715 

89515 

1.11713 

92709 

.07864 

96008 

.04158 

10 

51 

86470 

1.15647 

89567 

1.11648 

98783 

.07801 

96064 

.040U7 

!> 

52 

86521 

1.15579 

89620 

1.11582 

92817 

.07738 

99130 

.04036 

8 

53 

86572 

1.15511 

89672 

1.11517 

92872 

.07676 

96176 

.0397(5 

7 

54 

86623 

1.15443 

89725 

1.11452 

92926 

.07613 

W833 

.03915 

(5 

55 

86674 

1.15375 

89777 

1.11387 

92980 

.07550 

96288 

.03855 

5 

56 

86725 

1.15308 

89830 

1.11321 

93034 

.07487 

96344 

.03794 

4 

57 

86776 

1.1  5340 

89883 

i.narifi 

93088 

.07425 

96400 

.0373' 

3 

58 

8G827 

1  .  '5J72 

83935 

1.11191 

93143 

.073(5-2 

M457 

.03674 

2 

5i) 

86878 

i.  15104 

89988 

1.11126 

93197 

.07399 

96513 

1.03613 

1 

60 

86929 

1.15037 

90040 

1.11061 

93252 

.07237 

96569 

1.03553 

0 

M 

N  Cot 

N.  Tan. 

"N.  cot. 

N.Tan. 

~N.  Cot. 

N.  Tan. 

NTCoT. 

N.Tan. 

~M~ 

49  Degrees. 

48  Degrees. 

47  Degrees. 

46  Degrees. 

NATURAL  TAN&BNTS. 


44  Degrees. 

44  Degrees. 

M 

N.  Tan. 

N.Cot. 

M 

M 

N.  Tan. 

N.  Cot. 

M 

0 

96569 

.03553 

60 

"sT 

98327 

.01702 

29 

1 

96625 

.03493 

59 

32 

98384 

.01642 

28 

2 

96681 

.03433 

58 

33 

98441 

.01583 

27 

3 

96738 

.03372 

57 

34 

98499 

.01524 

26 

4 

96794 

.03312 

56 

35 

98556 

.01465 

25 

5 

96850 

.03252 

55 

36 

98613 

.01406 

24 

6 

96907 

.03192 

54 

37 

98671 

.01347 

23 

7 

96963 

.03132 

53 

38 

98728 

.01288 

22 

8 

97020 

.03072 

52 

39 

98786 

.01229 

21 

9 

97076 

.03012 

51 

40 

98843 

.01170 

20 

10 

97133 

.02952 

50 

41 

9890J 

.01112 

19 

11 

97189 

.02892 

49 

42 

98958 

.01053 

18 

12 

97246 

.02832 

48 

43 

99016 

.00994 

17 

13 

97302 

.02772 

47 

44 

99073 

.00935 

16 

14 

97359 

.02713 

46 

45 

99131 

.00876 

15 

15 

97416 

.02653 

45 

46 

99189 

.00818 

14 

16 

97472 

.02593 

44 

47 

99247 

.00759 

13 

17 

97529 

.02533 

43 

48 

99304 

.00701 

12 

18 

97586 

.02474 

42 

49 

99362 

.00642 

11 

19 

97643 

.02414 

41 

50 

99420 

.00583 

10 

20 

97700 

.02355 

40 

51 

99478 

.00525 

9 

21 

97756 

.02295 

39 

52 

99536 

.00467 

8 

22 

97813 

.02236 

38 

53 

99594 

.00408 

7 

23 

97870 

.02176 

37 

54 

99652 

.00350 

6 

24 

97927 

.02117 

36 

55 

99710 

.00291 

5 

25 

97984 

.02057 

35 

56 

99768 

.00233 

4 

26 

98041 

.01998 

34 

57 

99826 

.00175 

3 

27 

98098 

.01939 

33 

58 

99884 

1.00116 

2 

28 

98155 

.01879 

32 

59 

99942 

1.00058 

1 

29 

98213 

1.01820 

31 

60 

10000 

1.00000 

0 

30 

98270 

1.01761 

30 

M~ 

N.  Cot. 

N.  Tan. 

M~ 

~M~ 

N.  Cot. 

N.  Tan. 

"M 

45  Degrees, 

1 

45  Degrees. 

tl    /  /  x*4^C^>     (Let  f  •  ,    v/cWlo 


M     . 

**. 


